430 Prof. Donkin o/i the Geometrical Laias of the 



five instead of absolute, and consider the sphere there supposed 

 moveable to be fixed, attributing opposite rotations to the 

 sphere there supposed fixed, and we obtain the following 

 theorems : — 



Let ABC be anxf triangle on the moveable sphere. Then 

 twice the rotation BA followed by twice the rotation CB is 

 equivalent to twice the rotation CA. 



Referring to figure 1, and considering it as drawn on the 

 moveable sphere, we see that this is equivalent to the fol- 

 lowing proposition, viz. twice the rotation EB followed by 

 twice the rotation BD is equivalent to twice the rotation CA. 

 Whence it is obvious that we may enunciate the theorem as 

 follows : — 



Twice the rotation A.^ followed by twice the rotation BC is 

 equivaletit to twice the rotation DE. 



The second theorem will take the following form : — 



Let PQR be any tria7igle on the moveable sphere, the letters 

 being aiTanged as before. Then a negative rotation 2P round 

 V, followed by a negative rotation 2Q round Q, is equivalent to 

 a negative rotation 2(7r — R) round R. 



It is also easily seen that in this enunciation we may sub- 

 stitute positive for negative rotations, provided we adopt the 

 reverse arrangement of the letters P, Q, R, namely, that in 

 which Q shall be the negative pole of the rotation from QP 

 to QR. 



If in this theorem, whether referring to the fixed or move- 

 able sphere, we suppose the rotations indefinitely small, the 

 order of the two component rotations becomes indifferent, and 

 we easily obtain the ordinary propositions concerning the 

 composition of angular velocities. 



I shall not, however, enter into details upon this point, as 

 I wish at present to point out the use which may be made of 

 the preceding theory, as affording a direct connecting link 

 between the geometrical laws of the displacements of a rigid 

 system, and the analytical formulae of Sir W. Hamilton's 

 method of quaternions. In fact, the latter method furnishes 

 an algebraical theory of spherical trigonometry ; and we have 

 just seen that the geometrical laws in question depend in a 

 very simple manner on the properties of spherical triangles. 

 I must premise, however, that I have found it convenient to 

 use a system of interpretation of quaternions which differs, to 

 a certain extent, from that employed by Sir W. Hamilton 

 himself. It is possible, therefore, that the two following in- 

 stances of the application of the method may not be fully in- 

 telligible. I give them nevertheless, in the hope that I may 

 be allowed in a future communication to explain the prin- 



