174 AN EXAMINATION OF SOME QUESTIONS 



the sum of all the profits will be 



uP b + tf(P.)P* 



M P C + v(P a + PO P c 



uY d + v(P a + P» + Pc)P. 



uY e + v (P. -f P& + P c + Pi) P.. 



It is easily perceived that the co-efficient of u is equal to that 

 of the second term, and the co-efficient of v is equal to that of 

 thethird term of the equation 



+ P o ) 0+P 4 ) O + Pc) + P d ).. . rr 0; 

 but of the quantities * P OJ Pj, P c , . . p are equal to n l9 q are 

 equal to n fi , r are equal to w 3 , &c. this equation is there- 

 fore 



(x + n )p (x + n e )? (x + n s ) r . . . = 0; 

 and the co-efficient of the second term, in its developement, 

 is 



pn 1 .+ qn 2 -f rn s -f . . . 

 whilst that of the third term is 



+ T w i w 2 + + y n n 3 + q ~n 2 n 5 -j- . . . 



so 



* v This notation has been employed by Mr Herschel, in a paper in the Philo- 

 sophical Transactions for the year 1818 on circulating functions. 



