April 13, 1906.] 



SCIENCE. 



581 



These must be accounted for in any theoret- 

 ical solution, and an exhibit was made of cer- 

 tain formulae which are being tried to ac- 

 count for the observed velocity, pressure, tem- 

 perature and heat contents in the several 

 levels. 



The 615th meeting was held March 10, 1906. 



Mr. H. 0. Dickinson discussed ' Thermal 

 After-effects on Thermometer Glass,' describ- 

 ing experiments made at the Bureau of 

 Standards on new unseasoned tubes of Jena 

 glass. At about 400° C. glass is plastic to 

 internal strains while still rigid to external 

 strain.?. So by prolonged heating at high tem- 

 peratures the strains are relieved which have 

 been set up on manufacturing the instru- 

 ment and which cause a rise in the zero-point. 

 The Jena normal thermometer glass, which is 

 easy to work, is used up to 450° C; the Jena 

 borosilicon glass, which is very difficult to 

 work, is used up to 550° with an internal 

 pressure of twenty atmospheres. The results 

 were shown by lantern slides of curves that 

 indicated the change of zero as a function of 

 the temperature of annealing and the dura- 

 tion of exposure at this temperature. The 

 exposures were made in an electric furnace in 

 which the temperature was kept quite constant 

 for many days. 



Mr. E. A. Harris presented a paper entitled 

 ' On Function-Theory Analogues Relating 

 Chiefly to Mathematical Physics.' The chief 

 object of the communication was to show how 

 complex variables other than x -j- iy can be 

 utilized in spatial and physical problems. 



Since iCj, y^ in the equation 

 ^1 + iPi = e—i^{x -\- iy) 

 are coordinates of the point x, y referred to 

 axes 6 degrees in advance of the original axes, 

 it follows that x^, y^, z^, given below, are co- 

 ordinates of a point in space referred to a 

 new system defined by the Eulerian angles 

 <\>, and i/'. In the equations 



®i + iy-i + y«i = 6-** (x -{-iy) + jz, 



2/2 + ^2 -I- M = e-iO(yt + izi) + jx^, 



a's + *2/3 -|- js-i = e-*<l'{x2 + iy^) + ;»,, 



j, like i and 1, is a separating symbol; also 



i' = f^ — 1. By eliminating all quantities 



whose subscripts are 1 or 2, x^, y^, z, become 

 expressed in terms of x, y, z and the angles 

 (^, and \p. 

 The equation 



(a — ib) {x^ + iy-i) ^ {a + ih) (X + iy) 

 signifies that the point x-^, y^ is the point x, y 

 displaced by a rotation through an angle 6 

 where 



cos ^6 - 



sin §8 = 



b 



= a if 



■62 = 1, 



= 6 if ai' + fti' 



Similarly the equation 



(a — ib — jc) (x-i ■ 



■i2/i + ;%) = 



(a+b + jc) {x + iy- 



■ Ml 



if the i;-term be omitted from the products 

 as having no spatial interpretation, and if 

 a^ -\-V -\- c'^ 1, signifies that the point 

 x„ 2/j, 2j is the point x, y, z rotated about a 

 line in the 2/z-plane passing through the origin. 

 Again, the equation 



{a — -ib- 



■jc iji ijd) (a!i -t- it/i4- ;>i) = 



(a -\- ib -\- jo ± ijd) (x -j- ; 



+ ;"«), 



if the «j-terms be omitted from the products, 

 and if a^ -\-V -\- c' -\- dP = 1, is the general ex- 

 pression for a displacement by rotation about 

 a line passing through the origin. The quan- 

 tities a, h, c and d are essentially Eodrigues' 

 parameters and have the values 



a = cos % w, 6 = cos 7 sin 14 w, 



c = — cos p sin % w, d = cos a sin % a, 



a, p and y denoting the direction-angles of 

 the axis of rotation and <o the angle of dis- 

 placement. 



The use of a complex quantity was pointed 

 out in connection with expressions for the ac- 

 celeration when a moving particle is referred 

 to polar coordinates — the complex used being 

 a sort of extension of the ordinary symbol for 

 angulai* velocity. 



A semi-mechanical method of transform- 

 ing from one plane to another by means of 

 hyperbolic complexes of two dimensions was 

 outlined. An example of such transforma- 

 tion having a physical application is implied 

 in the equation 



