178 REPORT— 1898. 



n/5 + 1 

 coth u=z2l — 2_. 



2 

 53 When sinh u=2 



and cosh u=.<J ^ 

 cosech u-=z\ 



vo 



2 

 tanh ■2t= — ^— 

 n/5 



and coth ?t='^_ 

 2 



Sinh w'=| has the same values for the functions accordine to the 

 table in § 50. o " 



On the figure we have 



FK= v/5, i.e., the vertical reading at F is \/5. 

 GK=2 



The lines FG, HK, KH', G'F' are equal. 



54. To find u when sinh M=coth tt, we have 



M=sinh~^ %!, 

 = loge (sinh M + V sinh2?( + l) ; 



whence 



M=1'06 approximately. 



Conclusion. 



55. It may be noticed that just as some equations are more suited to 

 plotting by means of semi-logarithmic coordinates than by logarithmic 

 coordinates, so advantage may be gained by having only one, or two, of 

 the axes in three dimensions graduated logarithmically. 



The more frequent use of logarithmic geometry would tend to simplify 

 many theoretical investigations, and, apart from theory, the aid which the 

 method gives in computation seemed to be sufficient justification for the 

 publication of this paper. 



In conclusion I wish to thank Professor J. J. Thomson for several 

 valuable suggestions. 



Cavendish Laboratory, Cambridge. 



