Section III., 1903 [ 73 ] Trans. R. S. C. 



X. — Numerical values of certain functions involving e—^. 



By W. Lash Miller, Ph.D., and T. E. Eosebrugh, M.A. 



(Read May 19, 1903.) 



In dealing with certain problems of Chemical Kinetics,^ equations of 

 the form 



k du = ;5 ■ 



(^ _ 2)a (B — Z)^ (C — Z)y ... (1) 



are commonly met with. The tables published herewith were first com- 

 puted (to a less number of decimal places) in order to furnish a ready 

 means of integrating them. 



In Equation (1), A, B, 0, . . are positive constants, arranged in 

 order of magnitude, A being the smallest ; a, fj, y, . . are any real quan- 

 tities; and 2 may range from to any value less than A. 



In Equation (1) making the substitution 



Ay = A — z, (2) 



there follows 



i:^„ = -'iy 



in which 



y^ (1 -f by)l^ (1 + cyy . . (3) 



b = ^ c — ^ etc 



K = k A""-^ (B — A)l^ (^C—A)y... 

 Equation (3) may be written 



K du = . e-3!/ -I- '•y- — sîy= + ^y•'— . • • (4) 



where q =2/3 b, r = ^ 2/i b', s = ^ ^/i b\ t = ^ 2 /3 b*. 



If 6, c, . . , are small (y being obviously less than unity), the series 

 in the exponent of e converges rapidly, and Equation (4) may be 

 (approximately) replaced by 



Kdu = -^ . e-iy {I +ry'+h rh/ - sf + ty') (5) 



y 



Writing x for qy and integrating (with the lower limit m = 0, 2 = 0) 

 Ku = 



—Je-'''x~°- dx — ^fe~''x^-^ dx-^ ^fe''" x^^dx— '^} ^V e'-V-^^x (G) 



1 The subject is more fully discussed in a paper which will shortly be published 

 under the title : T. R. Rosebrugh and W. Lash Miller— A convenient integral form 

 of the Equations of Chemical Kinetics, 



