155 



40° C. at the Moon's equator, and by a small amount only at her 

 poles. This must be attended by an enormous monthly inequality, 

 amounting to nearly 60° C, supposing the matter of which her su- 

 perficial crust is composed to have the same conductivity, specific 

 heat, and radiating power as the crust of the Earth. If these be 

 much greater for the Moon, this inequality might be considerably 

 diminished. At the poles it must be comparatively small. 



The lunar temperatures here spoken of are those of the matter 

 forming her external crust. The temperature which would be indi- 

 cated by a thermometer placed in her immediate vicinity would be 

 affected by the Moon (in the assumed absence of an atmosphere) 

 only by her direct radiation. We have not the means of determining 

 what this temperature may be. 



Also a paper was read " On the singular Points of Curves." By 

 Professor De Morgan. 



Mr. De Morgan defines a curve as the collection of all points 

 whose co-ordinates satisfy a given equation ; and contends for this 

 definition as necessary in geometrical algebra, whatever limitation 

 may be imposed in algebraic geometry. He divides singular points 

 into points of singular position and points of singular curvature ; the 

 character of the former depending on the axes, but not that of the 

 latter. Both species are defined as possessing a notable property, 

 and such as no arc of the curve, however small, can have at all its 

 points. 



The form first considered is that of which the case usually taken 

 is an algebraic curve. Let <p(x, y) be a function which for all real 

 and finite values of x and y is real, finite, and univocal ; let the curve 

 be <p(x, y) = 0, considered as an individual of the family <p(x, y) = const. 

 The two curves d<p : dx=0, d<p : dy=0, or ^=0, <f> y =0, are the sub- 

 ordinates of this system, on which the singular points of all depend. 



When <p is not reducible to another function of the same kind by 

 extraction of a root, it divides the plane of co-ordinates into regions 

 in which, severally, it is always positive or always negative. By 

 this consideration it is easily shown (independently of y', y" , &c), that 

 if (x -f- dx, y + dy) be a point on the tangent at (x, y), <}>(x+dx, y -J- dy) 

 has the sign of <p X!C dx' i -\-2<f> X ydxdy-\-<pyydy' i . Hence, immediately after 

 leaving the curve, <j> agrees with or differs from — § y y" at the point 

 left, according as the curve is left on the convex or the concave side. 

 Hence easily follow the criteria of flexure, and also the following 

 relation between any two points whatsoever of the curve. 



Let two points be called similar when a line drawn from one 

 to the other cuts the curve an even number of times (0 included) 

 with the same abutments (on convexity or on concavity), or an odd 

 number of times with different abutments. Let other points be 

 called dissimilar. These points are similar or dissimilar, according 

 as their values of ty y . y" agree or differ in sign. 



An a priori proof is given that multiple points, cusps, and isolated 

 points, must be determined by ^=0, (p p =0, or can only take place 

 when both subordinates meet the curve. It is shown that, in the 



