1890.] to the Discussion of Laplace s Equation. 125 



f„ q and r be any three independent solutions of (4), then we 

 obtain a relation of the form 



ds = dp . P + dq . Q + dr . R. 



[Since the paper was read I have obtained an analogue for 

 another fundamental theorem concerning functions of a complex 

 variable, viz. the theorem that 



\f(z)dz=Q, 



I 



the integral being taken round a closed curve not involving any 

 singular points. 



If we integrate over a closed surface, we have 



I Uidydz +jdzdx + kdxdy) r = j J IdxdydzVr = 0. 



This theorem will of course require corrections similar to those 

 necessary to make the other theorem general, but the material 

 for furnishing these is ready to hand. It is to be noticed that 

 in the statement of this theorem, as in that of the theorem of 

 Article 3, the infinitesimal factor has to be placed before the 

 finite one. 



There is also a corresponding theorem for the case discussed 

 in Article 5, which may be written in the form 



I \\(dxdydz + idydzdt +jdzdxdt + kdxdy dt) s = 0.] 



(4) On a simple model to illustrate certain facts in Astronomy, 

 with a view to Navigation. By A. Sheridan Lea, Sc.D., Gonville 

 and Caius College. 



The model consists of a small solid sphere, representing the 

 earth, placed in the centre of a hollow sphere composed of circles 

 of wire. Of these, one represents the celestial equator, one the 

 ecliptic, and the others various meridians corresponding to the 

 parallels of longitude on the earth. Small coloured balls can be 

 attached at any point on the wire circles to represent at any time 

 the positioDs of the sun or any star relatively to the earth. A wire 

 representing the axis of the ecliptic can be attached to one of the 

 vertical meridians, and this carries the moon with the axis of her 

 orbit inclined at 5° to that of the ecliptic and movable round the 

 latter. The model of the earth is perforated by holes bored at 

 right angles to its surface by means of which a movable horizon, 

 carrying a wire at right angles to it which determines the zenith, 

 can be attached at any point of the earth. The model whi'e 



VOL. VII. pt. in. 11 



