Mr. J. W. L. Glaisher on a Class of Definite Integrals. 295 



title are devoted to the evaluation of Integrals by different iso- 

 lated methods, which, though well adapted for the purpose and 

 interesting, are not connected together as parts of a theory. A 

 similar want of system holds with regard to the integrals them- 

 selves. Many have been evaluated on account of their use in 

 physics, a greater number on account of their intrinsic interest 

 or elegance, more still as examples of different and frequently 

 highly ingenious modes of evaluation ; while in not a few cases 

 it is hard to see what inducement tempted their authors to 

 spend time over them. The subject of Definite Integrals,, re- 

 garded as a science, is still rather in the observational than the 

 theoretical state; i. e. the results have more resemblance to 

 detached facts observed than to a chain of facts connected by a 

 theory ; and so it must for a long time remain. 



Any one who takes up a memoir or work on the subject (such 

 as Meyer's Theorie der bestimmten Integrate, published during 

 the present year) must at once notice that, except in a few cases 

 such as the Elliptic Functions, the Gamma Function, &c, which 

 are usually considered separately, a number of definite integrals 

 are proved equal to certain quantities, without any indication 

 being apparent why those given should have been preferred to 

 others omitted, or, in the absence of any properties of the func- 

 tions, for what purpose they were evaluated. The fact seems to 

 be that every definite integral, not so complicated or unsymme- 

 trical in form as to be absolutely destitute of interest, which 

 admits of finite expression, or of expression in a tolerable simple 

 series, has been evaluated ; and the gaps which occur are due to 

 the integrals omitted not being so expressible. Thus, for ex- 



™P le >J o ^T^ = ¥a e ' but J -*£? " n0t eXpreS " 

 sible in terms of ordinary functions. The number of what may be 

 called principal integrals evaluable is not large ; and that this 

 is the case is not remarkable, when it is considered that there is 

 scarcely a function which cannot be thrown into the form of a 

 definite integral, while for the evaluation of the latter we can 

 only employ combinations of algebraical, circular, logarithmic, 

 and exponential quantities. For the advance of the subject, 

 therefore, the introduction of new fundamental functions is a 

 necessity ; and Schlomilch, in order to evaluate the second of the 

 integrals written above and some few others of allied form, made 

 use of the functions known as the sine-integral, cosine-integral, 

 and exponential-integral. As soon as it appeared that these were 

 suitable primary functions, a large number of definite integrals, 

 previously inexpressible, were reduced to dependence on them, 

 their properties were investigated, and Tables constructed of 

 their numerical values. 



