206 



as it may be written, P+ V — lQ=0 ; and the object is to show 

 that there will be at least one value of r between and positive in- 

 finity, and one value of d between and 2-rr, which, used in combi- 

 nation, will make both P and Q =0. 



This is effected by constructing two curves whose common ab- 

 scissa is 6, and whose ordinates are respectively the corresponding 

 values of P and Q, produced by substituting in their expressions the 

 same value of r, and observing the change which takes place in the 

 form of these curves, and in the position of their points of intersec- 

 tion, as r successively assumes all values from to positive infinity. 

 The existence of a root will be indicated by a point of intersection of 

 these curves (the P-curve and Q-curve, as they maybe called) falling 

 on the axis of abscissa?. When r=0, each of these curves will be a 

 straight line parallel to the axis of abscissae. When r=oo , the cor- 

 responding values of P and Q, will generally be indefinitely great ; 

 but by reducing their values in the same proportion, which will not 

 affect the validity of the demonstration, the Q-curve will become a 

 line of sines, and the P-curve a line of cosines, or a line of sines 



drawn back through — On constructing these curves, which we 



may call respectively P(0), Q(0), P(oo ), (Q oo ), it will be remarked — 



(1) That P(0) and Q(0) do not intersect. 



(2) That P(oo ) and Q(oo ) intersect in two points. 



(3) That one of these points of intersection is above the line of 

 abscissa?, and the other is below it. 



On considering the change in the forms of the P-curve and Q-curve 

 as r increases fromO to infinity, it will be seen that the P-curve must 

 have intruded on the Q-curve, at first by simple contact; and that, 

 as the intrusion advances, the simple contact is changed into two in- 

 tersections, which will at first be on the same side of the line of 

 abscissa?. But as, where r is indefinitely increased, any two conse- 

 cutive intersections necessarily lie on opposite sides of the line of 

 abscissa?, it may be shown, by considering the various ways in which 

 the intrusion may take place, that in all cases one at least of the 

 intersections must have crossed the line of abscissa? during the in- 

 crease of r ; and a root is thus determined. 



A communication was also made by Professor Miller " On the 

 contrivances employed by M. Porro in the construction of instru- 

 ments used in Surveying and Astronomy." 



February 14, 1859. 



Mr. Humphry made a communication " On the Limbs of Verte- 

 brate Animals." 



He gave a brief description of the fore and hind limbs in the 



