1874.] Adiabatics and Isothermals of Water. 455 



to any such point of intersection, but is only the common projection of 

 two separate points on the surface. 



In the first place, then, we know that if water, starting from an initial 

 state such that addition of heat at constant pressure is accompanied by 

 diminution of volume, be allowed to expand without receiving or emitting 

 heat, its temperature will rise ; i: e. it will at the same time be doing 

 work, solely at the expense of its internal energy, and rising in tempera- 

 ture a process which cannot go on indefinitely, as at last all the internal 

 energy would be due to the temperature alone, and any further per- 

 formance of work would necessarily involve a fall in temperature. 



-Hence there must be a point of maximum temperature on the complete 

 adiabatic drawn through the point representing the initial state ; and the 

 isothermals through all other points on the same curve which lie within 

 the region, in which addition of heat involves contraction, must meet it 

 twice. The projections of these curves will also necessarily intersect in 

 two points ; and since when an adiabatic and isothermal meet the 

 tangent to the former always makes the larger acute angle with the axis 

 of v (Maxwell, ' Theory of Heat,' p. 130), it follows that the two curves 

 must also cross at some point between their points of intersection, and 

 will thus form two loops. 



This result holds however near the points of intersection may be 

 together ; and when they coincide the curves on the characteristic surface 

 touch one another, and their projections on the plane of y>v have contact 

 of the second order, since three points, i. e. the two points of intersection 

 and the crossing point, are coincident ; and, further, the isothermal which 

 thus touches the adiabatic is evidently that which corresponds to the 

 maximum temperature above mentioned ; and the point of contact lies on 

 the curve which is the boundary between the regions in which elevation 

 and depression of temperature are respectively the results of compression, 

 for at neighbouring points on the adiabatic the temperature is lowered 

 when the volume is either increased or diminished. 



All the points of maximum temperature on the complete adiabatics lie 

 on the curve defined by the condition 



(D- 



and since at all points on this curve the tangent planes to the surface are 

 perpendicular to the plane of pv, therefore the projections on that plane 

 of all curves intersecting it touch its projection, because their tangents 

 lie in a plane perpendicular to that of pv 9 and are projected into one 

 line. 



Hence the projection of any curve which meets this curve must at the 

 projection of the point of section touch an adiabatic. 



But the ordinary interpretation put upon contact of an odd order with 

 an adiabatic is that the body passing through the cycle of operations 



YOL. XXII. 2 M 



