262 Mr. J. W. L. Glaisher's Numerical Values of Integrals. [Mar. 10, 



inches, the pressure perpendicular to the abdominal wall produced by the 

 action of the abdominal muscles was 

 1 13 



P= — — — =38*47 lbs. per square inch, 



2*93/ 



a result which differs little from that already found by calculation from the 

 actual measurements of the muscles and curvatures. 



II. n Tables of the Numerical Values of the Sine-integral, Cosine- 

 integral, and Exponential Integral." By J. W. L. Glaisher, 

 Trinity College, Cambridge. Communicated by Professor Cay- 

 ley, LL.D. Received February 10, 1870. 



(Abstract.) 



The integrals 



dll 3 1 (XUy I du y 



u | u \ u 



called the sine-integral, cosine-integral, and exponential integral, were used 

 by Schlomilch to express the values of several more complicated integrals, 

 and denoted by him thus, — Si.r, Ci>r, Ei#; the last function, however, is 

 for all real values of x only another form of the logarithm -integral, the re- 

 lation being 



Ei^lie*. 



These functions have since been shown to be the key to a very large class 

 of definite integrals, and several hundreds have been evaluated in terms of 

 them by Schlomilch, De Haan, &c, so that for some time they have been 

 considered primary functions of the integral calculus, and forms reduced to 

 dependence on them have been regarded as known. 



Considering, therefore, the large number of integrals dependent on them 

 for their evaluation, and their consequent importance as a means of extend- 

 ing the integral calculus, it seemed very desirable that they should be syste- 

 matically tabulated, the only values which have previously been obtained 

 being those of Si x, Ci x, Ei x 3 Ei (— x) for the values x= 1, 2, ... 10 calcu- 

 lated by Bretsehneider, and printed in the third volume of Grunert's 

 ■ Archiv der Mathematik und Physik,' and a Table of the logarithm-integral 

 published by Soldner at Munich in 1806. 



The present Tables contain the values of Si x y Ci x, Ei x, Ei (— x) for 

 values of x from 0 to 1 at intervals of *01 to nineteen places of decimals, 

 for values of x from 1 to 5 at intervals of *1, and from 5 to 15 at intervals 

 of unity, to ten places, and for j:=20 to twelve places. Also values of Si a? 

 and Ci a; only for values of j: from 20 to 1 00 at intervals of 5, to 200 at inter- 

 vals of 10, to 1000 at intervals of 100, and for several higher values to seven 

 places ; besides Tables of the maxima and minima .values of these func- 

 tions, corresponding in the case of the sine-integral to multiples of tt, and in 



the case of the cosine-integral to odd multiples of also to seven places. 



