174 AN EXAMINATION OF SOME QUESTIONS 



the sum of all the profits will be 

 uFa 



mP, + v(Pa + POPc 



UV, + v(Fa + Pi + Po)P<Z 



wPe + f(P. + P« + P. + Pd)Pe. 



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 



Gr-i-P.) Gt' + Pi) (-r + Pc) (a- + Prf)... zz 0; 

 but of the quantities * Pa, Pj, Pc, . . ^ are equal to n,, q are 

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

 fore 



(x + n )P (x + wj? (-J^ + n,)' • • • =0; 

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

 is 



pn, + qn^ ^ rn, -f . . . 

 whilst that of the third term is 



+ ^ n^n^+ ^ P^ n n, + ^n^n, + . . . 



so 



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

 ««iphical Transactions for the year 1818 on circulating functions. 



