DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
103 
of the function A will cause any of the new elements to be given functions of the old. 
But the problems most likely to occur may be solved in another way, as follows. 
35. Assuming - for the set ot l} a 2 , ... cc n , given functions of the set a x , a 2 , ... a n only, 
it is required to find j3 15 ... (3 n . 
(It will be observed that the conditions [a { , «,-]= 0 are necessarily satisfied in this 
case by virtue of (25.), art. 9, since a I5 &c. do not involve b x , &c.) 
It is plain, that if the principal function X had been found from the n integrals 
a x , a 2 , ... a n (as in art. 14.), it would be changed into that which would be found from 
the n integrals a x , a 2 , ... a„, merely by introducing the expressions for ... a n in 
terms of a 15 ... a n ; which expressions would be found by algebraical inversion of the 
assumed equations which give the latter set as functions of the former. Let X repre- 
sent the function X thus transformed ; we have then 
n dX. dX da x dX da 2 
d»i da x da-i' da . 2 dcti ' ‘ ' 
, da | j da<£ , da n 
:4 ‘s; +4 *s + 
(38. 
Thus (3 1 is determined as a function of the old elements, since &c. may be ex- 
pressed in terms of the latter. In like manner we should have a set of inverse equa- 
tions 
(39.) 
which may be used instead of (38.). 
It is apparent that /3 15 &c. will involve in general the elements &c. as well as 
b x , & c. 
Conversely, if we assumed for (3 1} &c. given functions of the set b x , ... b n alone, we 
to ( 19 .), but would not answer our present purpose. I regret to use symbols with a meaning different from 
that which custom has to some extent sanctioned ; but there seemed to be only a choice of difficulties. 
Mr. Spottiswoode has suggested to me the employment of the symbols (analogous to Mr. Sylvester’s 
“ umbral” notation) 
U, V, w, ... 
d d d 
> 
_dx’ dy dz' 
x, y, z, ... 1 
d x , d. 2 , d 3 ... J 
instead of those which I have used, namely. 
d(u, v, w, . . .) 
d(x, y, z, ...)’ 
d(x, y, z, ....). 
If these were adopted, the two forms ( p , q), [p, 9] might be used without confusion in their usual significations. 
See note to art. 9 . But although the “umbral” forms are more suggestive of the properties which belong to 
the above expressions as determinants, the other forms bring more into view the analogies which connect them 
with the differential calculus ; and therefore, for the purposes of this paper, I have preferred them. And it is 
perhaps better, for the present, that different notations should be tried, than that any attempt should be made 
to fix upon a definitive system for subjects so recent as those connected with the theory of determinants. 
