Demonstrations of certain Theorems 0/Xagrange # Laplace. 25 7 



, ,. . . d n u, 

 and, dtferent.at.ng, - = 



da? 



Now, by Case 1, ' U ^ =/ (#) ~^^> and b y supposition 

 " B ~* = f(yY~ l r^'> hence by substitution we obtain 



d n u 



From which we may now infer the truth of the lemma. 



Laplace' 's Theorem. If y = F [z + x .f (j/)] and u l = 

 (^); also if ?{*>)} = F^*) and/{F(^)} =/ x ( s ) ; then 



_ 

 the nth term being - d2 .-s T ' i .8.3..(-l) 



For by Maclaurin's theorem 2/! = U -f- . I . ; 



U X 



4 



term 



where U , , l , -^ |- &c. represent what ?/ L , and its dif- 



, , 



A XQ U XQ 



ferential coefficients become when x 0. 



Now, since y = F \_% 4- ^./(j/)]* .*. some function of y 

 must be = z -f- xf(y) ; we therefore have by the lemma 



- 



We may now evidently infer the truth of the theorem. 



Lagrange's Theorem. If y = z + xf(y) and 2/j = p ( 



then the preceding theorem becomes u\ 9(2) +/ ( 2 ) 



JV.& Vol. 9. No. 52. 4pr7 1831. 2 L XLV. 



