48 G. PLARR ON THE SOLUTION OF THE EQUATION Vp¢p=0. 
so that we have: 
—m =S9'ag’Bo'y 
—m,=3Sa¢'Bo’y 
—mM, = 2S¢'a.By=2Sad'a , 
where > represents the summation of the three terms obtained by the permuta- 
tion of a, B, y, in circular order, according to a, B, y, a, B, &c. 
If we consider the function / as the symbol of an operator on a linear 
vector-function ¢, we get 
J(() =P —m,9? +m,o—mM, 
and we know by HamiTon’s admirable theory that we have identically : 
S(¢(p)|=9. 
Like as in algebra we transform f(g) by the introduction of another variable 
in the place of g, so we may also transform /(¢) by the introduction of another 
linear vector-function in the place of ¢. 
We will introduce successively the functions a, €, defined by 
_ 1 A) ¢ . Ye) 
C= 73 ae? 7 

the differentiations being of course only symbolical of the operations. 
Thus we have for a: 
w=O-5. 
Once for all we put 
(5) Mm, =m, , 
Then 
(6) w= O—M,, O P=M,+a. 
This gives : 
; “mM 
J (m,+3)=fm,+af'm,+o° a sane 
Owing to fg =6g—6m,, we have /’(m,)=0, and we put 
(7) Pais Q=/m, : 
Thus we transform /(¢)=0 into 
(7 bis) {(m;,+)= Ae) =a + Po+Q=0. 
We may now, besides the values of P, Q, given by definition, form other 
expressions for them, analogous to those of m, m,, m2, either by deducing them 
from the independent consideration of the function a, or by the substitution of 
m;,+e, in the place of ¢ , in the equation #(¢)=0. By either method we shall 
arrive at the expressions following : 
