60 Prof. Jones on the Calculation of the Coefficient of 



we may express Po, Pi, Pg, &c. successively in terms of F (c) , 

 E(c), &c. The first three are : — 





2-0^ 2(l-c^ + 



2-3c2(i_c2)^ Sc^l-c^)'-^ • 



I have not expressed P3, P4, &c. in terms of F, B, &c., 

 because in practice it is easier to calculate them numerically 

 in any special case as follows, rather than from their expres- 

 sions in terms of F, E, &c. 

 (i.) Calculate 



^ rfQi c^^Qi c^^Qi 0^ 



or 



Qo Qi> Qij Qb &c. 



These may be successively calculated from the equation 



(l-c^)Qi=E 

 and its derivatives : — 



c(l-c2)Qi-2c2Qi=E-F, 



c(l-OQi + Qi(l-5r)-3cQi = 0, 



c(l-c^)Qi + 4(l-2c'0Qi-13cQi-3Qi=0, 

 &c. 



(ii.) Calculate Q2, Q3, Q4, &c. from the equations : — 

 Q2=Qi+|Qi, 



Q3=Q2+ K Q2==Q2+ ^ i ~Q I"qQiJJ 



n . r C '2A ■ 11 •• 1 ••• "» 



&c., 



which are obtained by successive applications of formula (III.)« 

 (iii.) Calculate Pg, P3, P4, &c. from these values of Q2; Qc, 

 Q4, &c. by successive applications of formula (II.). 



