( 372 ) 



I' n = Cr„ +2 



d.c = 



(*'-l) s 



71 + 3 



IdR'u 

 n dx 



(8) 



— ï 



from which, the R' n function being known, we easily derive the 

 expressions (10). Formula (8) holds good for all values of n except 

 n = 0, in which case : 



1' = 



r* 5 — 3 a-— 2 



(9) 



The A» coefficients, calculated by formula (6) being comparatively 

 large and the values computed by (8) small, it is desirable to omit 

 the factor (n +- 3) in (8) and (9) and to divide the expression for 

 A n by the same quantity; at the same time the sign of ,i, and there- 

 fore also the signs of (8) and (9), can be changed. 



With these premises the integrals assume the form : 



I = x (1_*') + 2(1 + x) 



i, = - (i-xy 



i.= 



i.— 



I \x 



X' — 



(10) 



i> 



— =J.U 8 - 



Xumerical values of these integrals calculated for values of x 

 increasing by 0.05 are given in Table IX; in the first column, which 

 remains the same for curves of different description, instead of 1 

 the product A a f has been given. 



Integrating formula (1) between the limits x and — 1, we encounter 

 the difficulty that, putting: 



x = 2z — 1 

 and developing, we obtain the form : 



n + 1 a + 2 a + 3 



6(6—1) i 



_,ƒ"* 



o + 6 + l 



x = 2 21 



a + 1 



— b 



a + 2 



a + 3 



— ett 



which, evidently, for large values of z slowly converges, so that a 

 considerable number of terms have to be taken into account if, as 

 is necessary in our case, we desire to determine the values with an 

 accuracy up to the third decimal. 



Other forms of development are of course possible, but 1 have 

 not succeeded in finding less laborious expressions. 



