INTRODUCTION. V 



/ (a), while the transformed series shall vary so slowly with reference to a, as to be 

 perfectly adapted to the ends already specified 1 



f (a) 



We should then tabulate 7,-V^ and, having taken a value of it from the Tables 

 for some value of a, it would only be necessary to multiply this tabular value by 

 the corresponding value of /'(«) to obtain the required value of /"(a). 



And if we could tabulate the logarithmic, instead of the numerical, values of 



I, ?l , we should find log /(«) at once, which is always needed. 



On these considerations the following Tables are based ; and, fortunately, only 

 slight and quite obvious changes in Leverrier's series were needed, thus, with corre- 

 sponding modifications, making the valuable labor of Mr. "Walker entirely available. 



9. Since we wish to tabulate the logarithmic values of irjK, it must be finite 



O f ( a )' 



for a = 0. 



But equation (2) fulfils this condition at once if we give it the form 



(3) ^ = 4, + i 1 « ! + i ! a ( + 4 3 a s + A t a 8 + ••••; 



for when a = 0, 



/(«) 



= A . 



Now, the rate at which log —^ is changing relatively to a, at any point between 

 the limits of a = and a = 0.75, is readily determined; and, if it be sufficiently 

 slow for practical use, then the series (3) needs no further change. 



Such was found to be the case with log — ±., which is tabulated on pages 1 to 7. 



10. When, however, as in most cases, the variation of the series (3) is too rapid 

 to be practical, it may be transformed hi the following manner. If we both multi- 

 ply and divide it by 1 — a 2 we shall obtain 



and by putting 



8? = , a „ , A l — ^0 = 8^0) A 2 — A 1 = SA 1 , A 3 — A 2 = 8A 2 , &c, 



1 — a* 



it becomes 



The whole variation of (3) from a = to a = 0.75 is 



(«) A(f) 2 + A(l) 4 +A(f) 6 + -"-; 



the whole variation of (4) between the same limits is 



(*J s A (t ) 2 + a A (I) 4 + s A (I ) 6 + ••••; 



the difference of these variations is 



(c) MIY + MiY + MiY + •••■• 



The coefficients in (3) are positive for all admissible values of s and i; and if 



