608 Miss P. M. Borthwick on Potential Measurements in 



neither more nor less than the standard formulae of Loba- 

 chevsky's imaginary or hyperbolic geometry for right-angled 

 triangles, the negative curvature being taken at unity. The 

 only difference is that for U(u) I have used Gr(u). 



Whether space has negative curvature or not *, Loba- 

 chevsky's functions are applicable to physical relations so 

 familiar as to be commonplace, and for this purpose they are 

 more advantageous than any others yet devised. 



In physical questions the gudermannian complement should 

 be employed for the most part rather than the gudermannian, 

 and these functions deserve to be introduced into elementary 

 mathematics t- 



Since Lobachevsky's IT(w) is equivalent to the guder- 

 mannian complement, a table of the gudermannian such as 

 that in ' Smithsonian Mathematical Tables ' becomes a table 

 of XI (w) by the simple process of subtracting the tabulated 

 values from tt/2, and this opens the way to the employment 

 of these functions in fields other than that of geology, notably 

 in electricity. 



Washington, D.C., 

 May 24, 1912. 



LVI. On Potential Measurements in the Neighbourhood of 

 the Electrodes in Point-plane Discharge. By Miss P. M. 

 Borthwick, B.Sc.% 



IN a paper on the ionizing processes at the surface of a 

 point discharging in air (Phil. Mag. Aug. 19.1.0), it was 

 shown by Drs. Ohattock and Tyndall that the value of the 

 field at the surface of a discharging point, under various 

 conditions, could be obtained by measuring the pull of the 

 lines of force on its end, which was made hemispherical for 

 the purpose. These field measurements were made by them, 

 and by Dr. Tyndall, for varying values of current, distance, 

 and size of point, both when the point was opposite a plate 

 and when it was subjected to a stream of ions arriving at its 

 end from an external source. These ions were of opposite 

 sign to that of the point and .are referred to below as 

 " external ions." 



* If the entity in which the properties of aether inhere is ponderable, 

 the sether must he confined to the portions of space characterized by 

 stellar bodies, the aether must be in a state of strain, and the path of 

 light is not rectilinear for Euclidian space. 



t For elementary work ''deformation angle'' might perhaps be a 

 better term than gudermannian complement. 



\ Communicated by Dr. A. M. Tyndall. 



