16. 



LAPLACE'S THEOREMS ON THE DEVELOPMENT OF FUNCTIONS 



IN SERIES. 



LET 



where P, Q, R are functions of x, y, z. It is required to develop a function 

 of x, y, z, in power's of a, b, c. 



We may prove that if D be the Jacobian 

 >?, Q = 1 I d(x, y, 



d(x, y, z) <:/( 77, ) 



dP ,dQ dR . d(Q, 

 = \-a-j -- b -j -c -j- + bc -f^- 



-j -- -j - -j- -- 

 dx ay dz d(y, z) 



d(a, b, c) 

 then for any function F ' (x, y, z) 



PF 



da \DJ d\ D )' db\DJ dy \ D ) ' dc \DJ ~ d\ D / ' 

 and generally 



JZ+m+n /S'X ,71+m+n 



da l db m dc n D1 d?dr m d n v D 



