DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
81 
If from this we subtract the corresponding equation obtained by inverting the order 
of the operations A, <5, remembering that A&m=&Aw, we obtain 
2 £ (^Ay«— + &a$b { )=0* (20.) 
(The use here made of the double operation AS, is due in principle to Mr. Boole. 
See his demonstration of a well-known theorem of Lagrange, of which the equation 
(20.) is a more general form-f-). 
If in this equation we suppose &r,-, by h &c. to be expressed in terms of Sa i5 hb t , &c. 
{Hyp. I.), and A a t , A b t , & c. in terms of Acr t -, A y { , &c. {Hyp. II.), and compare the terms 
on the two sides, it is easy to derive the relations (19.). I preferred however to 
deduce them by a more direct method. 
9. If x i be expressed in terms of the ‘In constants and t {Hyp. I.), and then each 
constant be expressed in terms of the variables {Hyp. II.), the result is an identical 
equation. Differentiating then with respect to x h x j} y k , we obtain the three equations 
dxi da x dxi db j dx t da 2 dx t db q 
rh r J”'db l dx{ da 2 dxi'db% dxi C ' 
' da x dxi 
dxi da 1 
' da x dxj 
0 — zzi daMdxi dbt 
"? - -7— . I rlh ri on . * H n /7/y». « rlh rl w . I 
dxi db x 
db x dxj 
da 2 dxj'db 2 dxj 
dxi da x dxi db x dx t da% dx t db z ^ 
da x dy k db x dy k da 2 dy k db 2 dy k ' 
Three similar equations may be obtained by treating^ in the same way. And if we 
apply to these six equations the transformations given by the system (19.), art. 7 , 
the resulting theorems may be comprehended in the following statement. 
If p, q be any two of the 2 n variables x x , ... x n , y x , ... y n , then 
/dp dq dp dq\_ /dbj da ; _ dbj doj \ 
l \dbi da t dai dbi) dq dq dp ) — ’ 01 
. . ( 21 .) 
according as p and q are or are not a conjugate pair , i.e. a pair of the form x j} y^ 
(The value +1 belongs to the case in which p=Xj, q—yj, and — 1 to the converse.) 
Here^? and q are a determinate pair of variables, and the summation refers to the 
constants, extending to the n conjugate pairs. 
More important however are the converse theorems obtained in a perfectly similar 
way by expressing a t , or b t in terms of the variables {Hyp. II.), and supposing the 
variables to be again expressed in terms of the constants and t {Hyp. I.). Differenti- 
ating the resulting identical equation with respect to a x , a j} b t , bj, and applying the 
transformations (19.) as before, we have, putting h, k for a determinate pair of con- 
stants. 
/ dh dk dh dk\ /dtp dx t dx t dy t \ 
* \dtji dxi dxi dyi) dh dk dh dk) — ’ 
* This might be written 
2i$(aa,yi) + HiS(ai,bi)=Q. 
See the notation proposed in art. 1. 
f Cambridge Mathematical Journal, vol. ii. p. 100. 
MDCCCLIV. M 
or =0 
■ • ( 22 .) 
