Mr. J. W. L. Glaisher on a Class of Definite Integrals. 431 



J» 4 .. : .. . 



agreeing with (59). 



It need scarcely be remarked that the function owes its im- 



portance in the Theory of Heat to the fact of u= Erf — -fr- 

 eeing an integral of 



du _, d*u 



dt ^doc 1 ' 



From its uses in physics^ it will be evident that the error- 

 function may fairly claim at present to rank in importance next 

 to the trigonometrical and logarithmic functions. 



Tables of the Error-function. 



For large values of x the formula (55) converges rapidly for 

 some distance, and affords a means of calculating Erf,r and 

 e x2 Erf^ with ease; and for small values the formula (53) gives 

 Erfca? conveniently. For intermediate values the three formulas. 

 (53), (54), and (55) are all inappropriate. It was this difficulty 

 of calculation which induced Kramp to compute his Tables. 

 Speaking of the series (53) and (55), he remarks : — " Les deux 

 series laissent done entr'elles un vuide pour devaluation exacte 

 de Pintegrale, qui peut aller depuis x — ^ jusqu'a a? = 4, et que 

 nous ne pouvons remplir par aucune des methodes connues. 

 Comme la connoissance de cette integrale est absolument essen- 

 tielle pour le calcul des refractions qui approchent de Fhorizon ; 

 comme elle est egalement indispensable dans P analyse de hasards ; 

 comme de plus la solution d'une infinite d'equations differentielles 

 revient a. cette meme integrale, j'ai cru qu'il valoit la peine d'en 

 calculer la table depuis a7 = 0'01 jusqu'a #=3*00" (Trait e des 

 Refractions, p. 134). The Tables which Kramp calculated are 

 three in number. Table I. gives Erf a? from <2? = to ,v = l'24s 

 to eight decimal places, from x= 1*24 to a?=l'50 to nine places, 

 from x = 1 "50 to x — 2*00 to ten places, from a? = 2 , 00 to a? =3*00 

 to eleven places, the intervals throughout being *01. Differ- 

 ences as far as A 3 are added from x=0 to ^=1*61, and as far 

 as A 4 from a?=l'61 to # = 3'00. Table II. contains log 10 (Erf#) 

 from x = to x = 3 - 00 at intervals of '01 to seven figures. First 

 and second differences are added. Table III. gives log ]0 (e* 2 Erf x) 

 from x—0 to x =3 "00 at intervals of *01 to seven figures. First 

 and second differences are added. Table I. was calculated by 



