PROBABILITY CURVES 105 



Substituting and equating coefficients of like powers of t 



Hence L 2 = (-/ 2 -2)/6, R x =-t/%, \ (U) 



Mi = 0, @ 2 = (' 2 -l)A f 



Starting with these initial values equations (8)-(10) are sufficient 

 to determine as many coefficients in the expansions (7) as are re- 

 quired. In order to demonstrate this, assume that all the coefficients 

 up to and including L k , Mk-\, -R/t-i. Qk have been determined. It can 

 then be shown that the next coefficient in each expansion can be 

 obtained from these data, as follows: 



For n = k-\-2, equation (9) can be written 



k 

 Q k+2 =L k+2 + 1 j±±L k+1 t+ JP k ^~ S L k+2 . s Q^^Q k+1 t, (12) 



5 = 2 



where Qk+i, Qk+2' L k+ i, L k+2 are the unknown quantities. For 

 n = k and n = k — \, equation (8) assumes the forms 



Rk = Qk+2-L k+ i + M k , (13) 



and R k -i = Qk+i-Lk+i+M k -i, (14) 



respectively, where all the quantities are unknown except Rk-i and 

 M k -\- For n =k, equation (10) can be written in the form 



L k+1 = R k +^^R k - s L' s+1 , (15) 



where L k ^ x and R k are the unknown quantities. Substituting in (12) 

 the value of (Q k+2 — L k+2 ) found from (13), and then substituting 

 the value of Rk found from (15) and the value of Qk+i from (14), 



LUi=Mk+-^^Lk+it+-j^(Rk~i+Lk+i-M k -i)t 



+Z 



k k—i 



k-\-2 — s T „ . -sr^ k — s 



L*+t-0.+ V ^-Rk-sL' s+ i. (16) 



k + 2 



,s = l 



This is a linear differential equation in L k +\ as a function of /, all the 

 coefficients being known functions of t with the exception of M k 

 which is an undetermined numerical constant. By a suitable choice 

 of the constant M k , (16) may be solved for L k +i as a polynomial in 



