()8 



right angles also passes through these points. Conceiving the step m. O A 

 drawn from A' we see that M A and A' M, kf^ M are the real and imagi- 

 nary components of the roots. The roots given by k/ and K'^ are by the 

 figure — 5 — 1.3m and — } -f- 1.3m. 



(c.) Real root of 2 x^ + 4 x- -f 8 -f 4 = o. 



We have O A = 2, AC = 4m, BC = 8m2= —8, CD = 4m3 = — 4m. 

 The circuit O A' W D was drawn by aid of transparent paper turned round 

 a pin with cross section paper underneath, after the manner of Lill's 

 wooden and ground glass discs. The root, A' A : m O A = tan k.' O A, 

 may be read ofT from the cross section paper to several decimal places. It 

 is here — .64.... 



O A^ B^ D is the circuit for the quadratic equation that gives the remain- 

 ing pair of roots of the cubic. The circle on D as diameter will not cut 

 A^ B' so that these roots are imaginary. 



On .soj[e theorems of ixtec;katioxs in qcatekxioxs. By A. S. Hatha- 

 way. 



There are certain identities among volume, surface and line integrals of a 

 quaternion function q=/(h) that include as special cases the well known 

 theorems of Green and Stokes, that are so often employed in mathematical 

 physics. These indentities were first demonstrated by Prof. Tait by the aid 

 of the physical principles usually employed in forming the so-called "Equa- 

 tion of Continuity." [See Tait's Quatermous, third ed., ch. XII J.] 



If dh dih,d2h be non-coplanar differentials of the vector h, the theorems 

 may be written : 



(1) — /fJSdhdihd2h.~q=/J V dhdjh.q 



(The surface integral extends over the boundary of the volume integral 

 and Vdhdih is an outward facing element of the surface.) 



(2) /fV (Vdhdih.~).q=/dhq 



(The line integral extends over the boundary of the surface integral in 

 the positive direction as given by the vector areas V dhdjh.) 

 These theorems are analogous to the elementary theorem, 



(3) /dq=qB — qj or in quaternion notation, 



•^ A 



— /Sdh'v.q=q 



