4.24 Mr. Herapath on the Integration 



the equivalent algebraic equations of the n — lth, ?? — 2th, . . 

 „_;; + ith dimensions. Therefore by this result and (11) 



-iR 



Pn-l=^n='n-X+/^n-x' [ "-'...(13) 



But according to the process we have followed 



xr 



y ^ pc 



xR 

 P = Pi^ 



xR, 



p,= p,e ' 

 • xR 



Pz = P3e ^ 



-.r(R + R +...R ) 



T, ,1 t — 1 t—2 t—v 



Ft ~ Pt-vC 



Now by the law of derivation which we have followed, Ro = R, 

 and Po = P', putting therefore v = t 



-x{R + R +...R) 

 p,=pe '-1 '-2 



x(R + R +...R+r) 



or, y—-pe=p^e . . . . ^I'l; 



a general expression for the value of y. Changing therefore 

 t into 11 — \ 



x(R +R +...R + r) x{R +R -t- ... R+r) 



y=Pn-x' "-';-' =X„e '^-^ "-3 (15) 



which since (12) gives a general relation between X , X _j 

 X^_2, ... is the integral of (7) the equation proposed. But 

 forasmuch as the quantities R, R,, Ro, . . . have no relation 

 assigned them, this solution can hardly be considered as prac- 

 tically serviceable ; and we shall hence endeavour to investi- 

 gate their general dependence on each other and on r. 



If the proposed equation (7) did not contain X, it is clear 

 that, since p e^ which is one value of y and would in that 

 case have p a constant, p^ would be a constant; and conse- 

 quently the exponent R^_j -f Rf_2 + . . . R + r must be a 



root 



