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



This procedure was truly scientific, and has extended the 

 limits of the science ; and a similar course must continue to be 

 pursued, not only with the view of increasing the number of in- 

 tegrals which, if need be, could be calculated numerically, but 

 also for the sake of making the subject more systematic and ho- 

 mogeneous in form, as well as connecting the different results 

 with more completeness and unity. 



The fact also, previously alluded to, of the power of definite 

 integrals as a means of expressing other functions (such as solu- 

 tions of algebraical and differential equations &c), points to the 

 value of a good classification accompanied by full numerical 

 Tables. 



The chief point of importance, therefore, is the choice of the 

 elementary functions ; and this is a work of some difficulty. One 



function, however, viz. the integral I e-^dx, well known for 



its use in physics, is so obviously suitable for the purpose, that, 

 with the exception of receiving a name and a fixed notation, it 

 may almost be said to have already become primary. I propose, 

 therefore, in the present communication to investigate some of 

 the most important integrals evaluable by its means, and several 

 connected results — and in a subsequent communication, after 

 noticing a few of the principal physical results involving it, to 

 describe the Tables that have been calculated of its numerical 

 values, and supplement them by a Table with different arguments, 

 which is nearly completed. 



As it is necessary that the function should have a name, and 

 as I do not know that any has been suggested, I propose to 

 call it the Error-function, on account of its earliest and still most 

 important use being in connexion with the theory of Probability, 

 and notably the theory of Errors, and to write 



Jm 



00 



e-*dx='Er{x (1) 



We then have the following results obtained by obvious trans- 

 formations : 



\ e-^ 2 doc=~^viax, (2) 



X 



e-"^Z = 4-™Vax, (3) 



Vx Va 



J 



\/ a 



e-^+^^=:_l_Erf(a? + 6)V«; ... (4) 



