THE VALUES OF -^fV'V//, 261 



terms outy will give correct results up to that point to nine places ; but at T = 2 

 (K= '822 656 449) the whole ten terms will be required to give eight figures correctly. 

 When T 2 consists of only two figures, the computation is easy, if we begin with the term 

 having the highest power of T. For the larger values of T, however, if not for all, it 

 is easier to derive the values of K by interpolation from those of H. 



7. It was a suspicion of some errors in the last figures of a few of the values in 

 these two tables in De Morgan's Essay, and in some values in Airy's Theory of 

 Errors of Observations (1861),* that led me to recompute the table of H. It was begun 

 during a holiday in the hot season of 1862 at an Indian hill sanatorium, where I had 

 very few books, and rather as an amusement to occupy the middle hours of the day, than 

 with any idea of publication. 



Commencing on a more extensive scale than Encke's table, in fact computing for 

 intervals of O'OOl, the values were worked out to about twelve places, but only nine 

 were preserved, together with first and second differences. To this I added the values t 



of -T^e" ( , partly as a check on the working, with differences. The work was at that 

 time advanced from t = to t = 1 '250, after which it was entirely laid aside for more than 

 thirty years. The computation of the portion carrying the argument to t= 3 is exceed- 

 ingly laborious, even with the intervals doubled after t = 1'5. But the values have been 

 given to fifteen decimal places from computations generally made to three or four figures 

 more, and might have been depended on as accurate even beyond the sixteenth place. 



This table, then, as recomputed, besides enabling us to construct Encke's second 

 table of K to seven or more decimal places, affords also the means of reconstructing or 

 verifying and extending Kramp's Table I. (for G) by means of the expression (4). 

 Several important constants also have been computed to a degree of accuracy perhaps 

 beyond any practical requirement. | 



The Formulae. 



8. The formulae available for computation, as pointed out by Laplace,§ are primarily 

 three, — (8), (10) and (11), with the continued fraction (13), which he supplied to facilitate 

 calculation where the series become very slowly convergent. 



(l) In the integral je~ t2 dt, if we develope e~ t2 , we get — 



r i fit 6 \ t 3 1 t 5 1 t 1 



/*(l-«'+ -p3+ etc.) =*- _ + .g--3- !T + etc., (7) 



* Op. cit., pp. 16, 20, 22-24. 



t I began by using the value of -?- given in Shortrede's Logarithmic Tables (1858), p. 602, viz., 1-283 791 670 946 99 



which is correct only to the tenth place, and therefore could not affect any of the results up to the eleventh place. 

 This was examined later, and the true value of the constant found to be 1-283 791670 955126. Shortrede's 



logarithm of j^ is correct. His value of sin 1° is also in error after the tenth decimal. 



X In the small table given by Airy, Theory of Errors, p. 24, six of the constants dependent on p are in error in 

 the 5th and 6th places, three of them in the 4th. 



§ Ttuorie Analytique des Probabilite's, 2e. ed. (1814), p. 103, and Me'canique Celeste, liv. x, c. i, sec. 5. 



