1. Introduction

The most common misuse of this expression is as an indication that the topic of discussion is changing. It cannot be used in this way. On the contrary, in general, its use implies the continuation of the topic of discussion. In particular, use of this expression is generally not appropriate to connect two statements about two different things, even when these things are closely related. Its misuse most frequently appears in connecting two such statements that describe somehow contrasting situations. This use is illustrated by the following.

(1^*) The solution ψ₁ is stable. On the other hand the solution ψ₂ is unstable.

(2^*) The solution ψ₁ is stable for β < 1. On the other hand, it is unstable for β > 1.

Note that the topic of discussion of the first sentence in (1^*) is the stability of ψ₁, and that of the second sentence is the stability of ψ₂. The sentences in (2^*) contrast two closely related cases, but here too the topic changes: That of the first sentence is the stability below β = 1, and that of the second sentence is the stability above β = 1. In these examples, the assertions of the first and second sentences describe contrasting situations, but they cannot be regarded as presenting opposing views, since the topic of each is different. Depending on the intended implication, ``on the other hand" here should be replaced by ``and," ``but," ``while," or something similar in the following manner: ``The solution ψ₁ is stable, and the solution ψ₂ is unstable."

The appropriate use of ``on the other hand" is demonstrated by the example below.

(3) The first method does not involve such a complicated integration, and for this reason it is usually more practical. On the other hand, the second method never produces unphysical solutions, and therefore it is more reliable.

Here, while the first sentence discusses the first method and the second sentence the second method, the main topic of discussion of both sentences is the relative utilities of the methods. With regard to this single main topic, these two sentences provide opposing points of view.

Below, I give a number of examples demonstrating the most common types of misuse of ``on the other hand."

2. Some simple examples:
The pattern ``A is/does... On the other hand B is/does..."

When we make two statements about the identity, nature, behavior, etc., of two different things, even if these things are closely related and the statements present some kind of contrast, we generally do not use ``on the other hand." The following are some fairly simple examples of this type of misuse.

(1^*) Here s_mn is the traceless part of h_mn. On the other hand t_mn is the remaining part, h_mn - s_mn.

(1) Here s_mn is the traceless part of h_mn, and t_mn is the remaining part, h_mn - s_mn.

(2^*) Here, F^L_{μ ν} and F^R_{μ ν} are local field strengths; on the other hand, F_{η η} and F_{μ η} are bi-local field strengths.

(2) Here, F^L_{μ ν} and F^R_{μ ν} are local field strengths, and F_{η η} and F_{μ η} are bi-local field strengths.

(3^*) It is thus obvious that by taking the limit χ → χ _μ, the extended derivative operator D^R+_μ tends to the local operator ∂⁺_μ. On the other hand, D^R-_μ tends to -ig∂^-_μ.

(3) It is thus obvious that by taking the limit χ → χ _μ, the extended derivative operator D^R+_μ tends to the local operator ∂⁺_μ, while D^R-_μ tends to -ig∂^-_μ.

(4^*) Player 1 succeeded in constructing a productive game environment, while player 2 failed to do so.

(4) Player 1 succeeded in constructing a productive game environment, while player 2 failed to do so.

(4^') Player 1 succeeded in constructing a productive game environment, but player 2 failed to do so.

(5^*) Under this extension, the Higgs-like fields in this model become bi-local. On the other hand, the gauge fields and the matter fields remain local, existing in either the left or right world.

(5) Under this extension, the Higgs-like fields in this model become bi-local, /although/while/but/ the gauge fields and the matter fields remain local, existing in either the left or right world.

(5^') Under this extension, the Higgs-like fields in this model become bi-local. /However/By contrast/, the gauge fields and the matter fields remain local, existing in either the left or right world.

(6^*) In (a), the average value only with respect to the periodic part is plotted. On the other hand, the average value including the transient part is plotted in (b).

(6) In (a), the average value only with respect to the periodic part is plotted, whereas in (b), the average value including the transient part is plotted.

(7^*) Here, the subscript denotes the multiplicity of the spin states, 2 for a spin-doublet and 4 for a spin-quartet. On the other hand, the superscript denotes the degeneracy of the SU(3) flavor state, 8 for a flavor-octet and 10 for a flavor-decuplet.

(7) Here, the subscript denotes the multiplicity of the spin states, 2 for a spin-doublet and 4 for a spin-quartet, and the superscript denotes the degeneracy of the SU(3) flavor state, 8 for a flavor-octet and 10 for a flavor-decuplet.

In each of the above examples, while the first and second sentences describe some kind of contrasting situations, there is no opposition represented by this contrast because the topics of discussion differ.

3. Some more complicated examples

(1^*) Since this theorem concerns an analytic function defined on a Riemannian surface, the metric signature appropriate for this manifold is Euclidean. On the other hand, the background manifold which we have considered in this paper has Lorentzian signature.

(1) Since this theorem concerns an analytic function defined on a Riemannian surface, the metric signature appropriate for this manifold is Euclidean. However, the background manifold which we have considered in this paper has Lorentzian signature.

There is certainly a contrast expressed here with regard to the type of metric signature considered in different situations. However, the problem with (1^*) is that these are different situations. Certainly the fact that the author considered a Lorentzian signature does not oppose the fact that the Euclidean signature is appropriate in other situations. (Even in the case that the intended meaning is that a Euclidean signature is in fact appropriate in the situation considered in the present paper -- and perhaps a Lorentzian signature was inappropriately used -- ``on the other hand" is incorrect. The fact that the Euclidean signature is appropriate and the fact that the author used the Lorentzian signature are not in opposition. Taken together, they simply imply that he used the wrong signature.)

(2^*) We thus see that the irreversible work associated with both the loose regime and the tight regime can be made as small as we wish by allotting a large enough time for the operation. On the other hand, the quasi-static work associated with the change of ν within the region χ_{α 0} < χ_α < χ_{α 1} can be evaluated by Eq. (3.1).

(2) We thus see that the irreversible work associated with both the loose regime and the tight regime can be made as small as we wish by allotting a large enough time for the operation. The quasi-static work associated with the change of ν within the region χ_{α 0} < χ_α < χ_{α 1} can be evaluated by Eq. (3.1).

There is clearly no opposing points of view here, since the two sentences consider completely different things.

(3^*) The player of this species usually cuts tree 1 for several successive rounds, and as a result this tree becomes shorter and shorter, as the amount of lumber the player acquires each round becomes less and less. On the other hand, tree 2 becomes taller and taller.

(3)The player of this species usually cuts tree 1 for several successive rounds, and as a result this tree becomes shorter and shorter, as the amount of lumber the player acquires each round becomes less and less. Meanwhile, tree 2 becomes taller and taller.

Again, these sentences do not in any way express opposing points of view.

(4^*) Using the conventional method, the evolution of the decision making function itself cannot be investigated. On the other hand, this evolution is systematically investigated using the S diagram in our method.

(4) Using the conventional method, the evolution of the decision making function itself cannot be investigated. Using our method, by contrast, this evolution can be systematically investigated, and this is done using the S diagram.

Note that the two sentences here not only are non-opposing but, in fact, they both support the conclusion that ``our method" is superior.

(5^*) The solution ξ₊(φ) is the only defect solution that satisfies the boundary conditions. This solution becomes identically zero for d ≤ 2, and thus no defects exist in in this case. On the other hand, this solution represents a defect for d > 2.

(5) The solution ξ₊(φ) is the only defect solution that satisfies the boundary conditions. This solution becomes identically zero for d ≤ 2, and thus no defects exist in this case. For d > 2, however, this solution is non-zero. Thus in this regime the system possesses a defect.

The contrast presented here is that the defect solution is zero in one case and non-zero in the other. [This is made somewhat unclear by the wording of (2^*).] Obviously, these assertions are not in opposition. In the sentence before ``on the other hand," the topic of discussion is the nature of this solution and its implications in the d ≤ 2 case, while in the sentences following this expression, it is the nature of this solution and its implications in the d > 2 case.

(6^*) It should be noted that only symmetric fluctuations exist initially. On the other hand, the inflaton itself generates asymmetric fluctuations.

(6) It should be noted that only symmetric fluctuations exist initially. The asymmetric fluctuations are entirely generated by the inflaton itself.

The two sentences here express no contrast. In fact their meanings are almost the same.

(7^*) This relation yields α < 2 γ. On the other hand, from (3.4), we have α < 2 π.

(7) This relation yields α < 2 γ. In addition, from (3.4), we have α < 2 π.

Here, the second sentence does not present contrasting information but, rather, additional information.

(8^*) The distribution of the average scores is quite smooth if we consider only the attractor part of the dynamics. On the other hand, if we included transient behavior in calculating the average score, when the number of rounds is small this distribution is quite irregular.

(8) The distribution of the average scores is quite smooth if we consider only the attractor part of the dynamics. However, if we included transient behavior in calculating the average score, when the number of rounds is small this distribution is quite irregular.

Here the contrast is between the two different things, the distributions in the two cases.

(9^*) In this way an attractor can be defined topologically. On the other hand, Milnor (1985) defined an attractor from another viewpoint in which both topological and measure-theoretic concepts are taken into account.

(9) In this way an attractor can be defined topologically. However, there are other ways to define attractors. For example, employing another point of view, Milnor (1985) defined an attractor in terms of both topological and measure-theoretic concepts.

There is nothing even contrasting expressed by the assertions before and after ``on the other hand" in this case.

(10^*) Thermodynamics, which is the study of heat, has been studied for many years, and a number of relations have been derived concerning the flow of heat and its conversion into other forms of energy. On the other hand, Brownian motion has also been studied for many years, and projection methods developed in this study have allowed for the derivation of Langevin dynamics from microscopic Hamiltonian mechanics.

(10) Thermodynamics, which is the study of heat, has been investigated for many years, and a number of relations have been derived concerning the flow of heat and its conversion into other forms of energy. Brownian motion has also been studied for many years, and projection methods developed in this study have allowed for the derivation of Langevin dynamics from microscopic Hamiltonian mechanics.

Again, there is no real contrast presented by the statements before and after ``on the other hand." The topic of discussion simply changes. Of course, the two types of study discussed here are in some sense contrasting, but the assertions of the two sentences themselves express no opposition. Taken together, they simply imply that two different approaches to the investigation of a particular class of physical phenomena have been used.

(11^*) When 0 < α < α₀, the equilibration time is sufficiently long that the change in the small system can be thought of as quasi-static. On the other hand, the regime α₁ < α < 1 corresponds to the case in which the equilibration time is much shorter than what can be discerned experimentally.

(11) For 0 < α < α₀, the equilibration time is sufficiently long that the change in the small system can be thought of as quasi-static. As the other extreme, for α₁ < α < 1 the equilibration time is much shorter than what can be discerned experimentally.

Here there is contrast between the statements before and after ``on the other hand," but, again, the topic of discussion changes. The first sentence is with regard to the small α regime and the second sentence is with regard to the large α regime. Clearly, assertions regarding the equilibriation time in these two independent regimes cannot be in opposition.

(12^*) Since the late 1940s, there have been many theoretical and experimental investigations of such systems at very low temperatures, and their behavior in this regime is fairly well understood. On the other hand, in 1992, employing a novel calculational technique, Allison and Carew carried out the first systematic investigation of their `high-temperature' behavior.

(12) Since the late 1940s, there have been many theoretical and experimental investigations of such systems at very low temperatures, and their behavior in this regime is fairly well understood. Then, in 1992, employing a novel calculational technique, Allison and Carew carried out the first systematic investigation of their `high-temperature' behavior.

Again there is clearly nothing expressed by the second sentence that opposes the assertions of the first sentence.

(13^*) The work required to change the parameter a during the quasi-static adiabatic process A → B is less than W_h. On the other hand, we can also consider the process B → A.

(13) The work required to change the parameter a during the quasi-static adiabatic process A → B is less than W_h. Of course, we can also consider the process B → A.

Here the ideas expressed by the two sentences are not even contrasting.

4. Unclear opposition

(1^*) Since the interaction between a vortex and the insect's wing is simple, the lift generated by a single vortex is essentially symmetric. On the other hand, the long lifetime of vortices implies that vortices produced in different downstrokes may interact.

(1) Since the interaction between a vortex and the insect's wing is simple, the lift generated by a single vortex is essentially symmetric. However, the long lifetime of vortices implies that vortices produced in different downstrokes may interact.

(2^*) Because the T-duality group G is a subgroup of the U-duality group it has a special property: It is the maximum subgroup which consists of the elements that transform H-S and Q-Y fields into themselves. On the other hand, we often encounter situations in which H-S and Q-Y fields are better treated in a different way.

(2) Because the T-duality group G is a subgroup of the U-duality group it has a special property: It is the maximum subgroup which consists of the elements that transform H-S and Q-Y fields into themselves. However, while this property is convenient in some situations, we often encounter situations in which H-S and Q-Y fields are better treated in a different way.

There seems to be something of an implied opposition between the first and second sentences in (1^*) with regard to the question of whether or not the lift experienced by the insect is symmetric. However, in fact, neither sentence gives clear support for either conclusion, and thus they cannot be regarded as expressing opposing points of view. In (2^*), the sentences may be viewed as expressing opposing views with regard to the use of the T-duality group in the treatment of these fields. However, while the first sentence seems to express a merit of its use, the second sentence clearly does not express a demerit.

5. Proper uses of ``on the other hand"

(1) The T^*-product is certainly very convenient, because we need not be concerned with the order of operation, even for field operators that are separated in a time-like manner. On the other hand, as emphasized in the present paper, the T^*-product can also create serious problems involving the explicit violation of the field equations.

(2) This treatment ignores the difference in symmetry of the two systems. On the other hand, our main interest here is in certain aspects of the macroscopic behavior that are independent of the symmetry.

(3) When an unproductive wheat field is left fallow for a sufficiently long time, there is a possibility that it will regain its productivity. On the other hand, if the field is left fallow for too long, long-range productivity will inevitably decrease.

The topic of discussion in (1) is the utility of the T$^*$-product. The first sentence clearly expresses a merit of its use, while the second sentence clearly expresses a demerit. The discussion of (2) is with regard to the appropriateness of the treatment in question. The first sentence raises a question concerning its appropriateness, while the second gives reason to disregard this question. (3) regards the advantage of leaving a field fallow. The first sentence states what can be gained by doing this, and the second states what can be lost.