354 
PROFESSOR DONKIN ON THE 
but cot ^ COS (p, and therefore 
d^dct 
= cot ^ sin ^ COS <p ; 
dH 
and in like manner we find the same value for so that in this particular case 
dudp 
the condition ^= 5 ^; is verified, 
dH dH 
But if we take and in the same way, we find the former = sin <p and the 
latter = 0 , so that in this case the condition is not verified. The geometrical meaning 
of this is obvious ; analytically it is merely an instance of a general fact, [tointed out 
by Jacobi ; namely, that the effect of two successive pseudo-differentiations with 
respect to two independent variables, is not generally independent of the order of 
operations. 
If V be the potential of another body, given in position, upon the body considered, 
then V is a function of <p, and 
dY , dY, , dY 
and if we substitute for dd, dp, d\p their values in terms of da, d[3, dy, we obtain an 
expression which we may call Lc^a+Mc?/3+N</7, L, M, N being functions of 0, p, ■<}/, 
of which, as is well known, the mechanical meanings are the moments of 
dS flip dp 
the attraction of the second body about the three axes. Here again no one would 
dY 
maintain that V is a function of a, (3, y ; and if, as is often done, we say -^ = L, &c., 
the above remarks apply in all respects to these equations. 
I should have thought it superfluous to dwell so much on these points if it had not 
appeared that writers on physical astronomy have in some instances either overlooked 
the distinction between real and jo,seMfi/o-differentiation, or at least have failed to point 
it out to their readers. The only discussion of the subject which I have met with is 
that given by Jacobi, in the correspondence referred to. 
It may be added, that in general investigations, where symbols such as &c. 
may be used without defining the nature of V, or the precise meaning of a, (3, &c., 
dW d^Y 
dctd^ d^da. 
d^Y d^Y 
serious errors might be committed if it were assumed that the condition 
always subsisted. 
Appendix C. 
The theorems relating to the transformation of coordinates, given in Section VI., 
may be made more general, and in many cases more useful, as follows : — 
