123 Mr. A. Cayley on an Analytical Theorem relating to 



where on the right-hand side the finite series is to be continued 

 lip to its last terra. And the equation hokls for any integer 

 value of i which is 2 . This is the simplest form of Plana's 

 theorem. 



We have ©^=2(1 + Z')-^G.-i ; 



, n 



or writing this equation under the form @i=i{l + b) tttvj fi"^ 



comparing with Plana's developed expressions for ^|~^ (which 



A." 



are continued by liim as far as G17), we find 



©4 = ^% 



0, =-^A- b^ -p^ + p-" + b^+ lb% 



©8 = p^ + 453 +^b'+ 465+ I b% 



,-, 3 , 13,0 32,0 . 24,. 24,. 32,/; 12,7 ^ in 



©10= -^b^-l2b^ -S7b^-54>b^-B7b<^-l2b'' ~^b^, 



&c., 



which are of course the results obtained by developing the fore- 

 going expression for 0,-, in powers of b, and collecting the terms. 

 The formulae put in evidence a remarkable symmetry which does 

 not exist in the original expression in powers of {1+b). 



It would be now easy to verify, for moderately small values of 

 the suffix, the equations 



3^ 3 1 



7;2©5+p 



&c. 



®3+T04+T5 ©5+7306 = 0, 



This is, in fact, Plana's process, which, however, as the suffixes 



