300 proceedings: philosophical society 



Vm=V(^6) = i(a + 6)|l - jj^p> (1) 



In this m is any number and a and b are any two numbers whose pro- 

 duct is m. Attention was called to the rapid approximation which 

 may be secured by successive applications of the arithmetic mean 

 \{a + b). Special emphasis was given to the use of formula (1) when 

 m is an integer and not a square. In this case the square root of m is 

 involved in the two equations 



y 2 — mx 2 = ix 

 y 2 + mx 1 = v 



in which every symbol represents an integer. These equations give 1 



IH-'-^L.-^*- (2) 



2xy 4 xy v 16 xyp 3 



The number ju is arbitrary within certain limits. For the present pur- 

 pose it is obviously most advantageous to have y. = + 1, and it is known 

 from Fermat's theorem that n = +1 is always possible. Moreover, 

 an infinite series of sets of values of x and y exists, each set satisfying 

 the equation y 2 — mx 2 = +1. Hence an infinite series of increasingly 

 rapid approximations to the y/m is furnished by equation (2). 



The paper was discussed by Messrs. Wead, Farqtjhar, Van Os- 

 trand, and Harris, particularly with reference to earlier methods. 



Mr. W. W. Fraser then presented a communication on Vectors and 

 quaternions; what has been done and what can be done. Among his defini- 

 tions we find that Hamilton has defined the quaternion as the quotient 



of two vectors a, /3, as q = - ; and as a set of four, or q = x -f- iy x + jyi 



a 



-+- ky 3 where i f j, and k are algebraic extraordinaries such that i 7 = 

 j2 = fca = —l } ij = k= — ji, etc., making the quaternion analysis an 

 algebra of sets in which the commutative law for the multiplication of 

 the extraordinaries is thrown out. Grassmann's Ausdehnungslehr dif- 

 fers chiefly from the quaternions in his definition of the product of two 

 vectors, being defined (inner), A.B = \A\\B\ cos (A,B); and (outer) 

 A X B = n | A|| B \ sin (A,B) (after Gibbs) : The Borali-Forti assump- 

 tion that i X ( ) = i (of Hamilton) = V — 1 affords a means of unit- 

 ing the systems of Grassmann, Gibbs, and Hamilton, since we can effect 

 translations from scalars, rotations with complexes or quaternions, and 

 projective transformations with the dyadic of Gibbs. If Ohm's law for 

 alternating currents is expressed with Grassmann vectors instead of 

 complexes, as used by Dr. Steinmetz, the difficulties of the latter's sym- 

 bolic method are avoided. 



Mr. W. J. Spillman then presented the results of an investigation in 

 collaboration with Messrs. H. R. Tolley, and W. G. Reed on A gra- 



ft 7)1 X 



1 Equation (2) may be derived from (1) by writing a = - and b = — . 



x y 



