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DEPARTMENT OP PURE MATHEMATICS. 483 



Uue greut iidvantage of tho quateruion uotatiou over all others is that it cau 

 be applied at once, without further definitions, to products of more than two vectors. 

 The non-associative vector algebras cannot treat of vector products until the 

 Tectors have been grouped in pairs. The two apparent exceptions given above 

 are really cases of contracted symbolism. 



Heaviside's notation difl'ers from Hamilton's in dropping the S before the 

 scalar of a product and changing the sign ; but here also the notation is inapplic- 

 able to higher products, because by putting the square of a unit vector equal to + 1 

 Heaviside makes his system also non-associative. 



It has been said that the quaternion is not required in physical applications. 

 In a certain sense this may be admitted ; and yet in no other vector algebra can 

 finite rotations be treated as they are in the quaternion system by means of the 

 beautiful operator g { )!Z ~ '• But as a matter of strict logic the quaternion product 

 of two or more vectors is always present. Its fundamental properties determine 

 the whole method of the calculus. Certain parts of it come to the front inces- 

 santly, just as in ordinary trigonometry the sines and cosines have an analytical 

 importance far exceeding that of the angle or arc. 



MONDAY, AUG VST G. 

 Departmbnt of Pure Mathematics. 

 The following Papers were read :•— 



1. JEnpdnsioiis in Products of Oscillating Functions, 

 liy Professor A. C. Dixon, F.R.S. 



The problem is to prove the validity of the expansion of a function of two 

 variables, x, ij, in the form 



2A</.(.r)^|.(y), 



\yhere <^ (.i), -^ {tj) are functions satisfying certain difl'ereutia] equations, 



'*^!^=Pd),'^ = Qx^, P, Q are functions of x, y respectively, and also linear 

 dx^ dy- 



functions of two parameters, X, /z. The summation extends over the pairs of values of 

 X-, fi given by such a system of equations as </> (a) :s (^ (6) = = >//■ {a)\\r {b), where a, b 

 ai-e the ends of the range of real values over which the expansion is to hold good. 



The investigation depends on an expression for ^ (f> {x) ^ (t) -^ (y) -^ {u) in 

 the form of a double contour integral. 



2. Anemoids, By Professor W. H. H. Hudson, M.A. 



An Anemotd is the path of a particle of air in a storm on the following 

 hypotheses : — • 



(1) The centre (Q) of the storm moves in & straight line. 



(2) The direction of motion of the air particle (P) makes a constant angle 

 (i „-a) with the join of the air particle to the storm-centre. 



(3) The velocity of the storm-centre has a constant ratio (/x) to the velocity of. 

 the air-particle. 



(4) The motion is in one plaice. 



Takmg rectangular axes as indicated in the diagram, in which PT, PG are 

 tangent and normal to the Anemoid, we have 



~ {x + y tan (</> — a)} ^ n 

 as ' 



112 



