OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
491 
exp + Mo|77 + . . . + + . . .) 
= 1 + mio£ + + • • • + + . . ., 
■where r) are the undetermined algebraic quantities. 
For the multiplication of two operators we have the formula 
9pi% — 9ppix 9 pa 9piqi "j~ 9pi<ii^ 
wherein the symbol “[■ denotes explicit operation upon the operand, regarding the 
latter as a function of symbols of quantity only, and not of the differential inverses. 
Also 
9piqi "j" 9pi'h ~ 9pi+ppqi+q2' 
Put 
"“i = + '^H\9i)\ + • • • + '^hiq9pq + • • 
which may be written 
tq = + • . . + + . . .) p, 
in which is .symbolically written mp,g. 
But 
— tq -j- ^q = (??qQ + + . . . + + . . .fg. 
where, after expansion of the right hand side, is to be written 
'^'^Piq^^y/h9pi+P2, ?i+?2- 
Then, with a similar convention, 
tq = tq -j- iq_i = q. ^0^ -f . . . + + . . .)V/. 
Further, it is easy to prove the relation 
Ug -j- 111 + (• 
But for a series of linear partial differential operators enjoying this property. It is a 
well known and easily established theorem of Sylvester’s that 
exp ?q = exp (^q — i i Rg — . . . 
Hence, substituting, we easily reach the relation 
exp (M^o^ + ^ 01 ^ + . . . + + *..) = ! + wqo^+ + .. . + + ■ • • , 
\ 
wherein ^ and rj are undetermined algebraic quantities. 
This establishes the theorem. 
3 R 2 
