OP ARTS AND scif:nces. 139 



common, or else a minimum of v^ or v.^ corresponfls with a maximum 

 of ^2 or Vy Here special investigation is necessary for each particu- 

 lar case. 



Observe carefully that, if the functions v, v^ and v.^ attained true 

 maxima or minima at the hot points, instead of coming to a point, the 

 proposition would not be true. 



That the proposition is true, in regard to shooting points, can be seen 



thus : ^ = tan r ; tan r changes sign instantaneously at a shooting 



point, and hence, for a small (infinitely small) change in a:[c?2:], ^ 



has a finite change of value, and d^-j- = finite, dx = ^. There- 

 fore 



d^v finite 



^^dJ = 1 = CO 



dx 



This is the condition for a shooting point. If, at the same point, -j- 

 d:?o 



as well as --— is infinite, there is a cusp. If also t? = oo , there 



13 



an asymptote. 



o 1 • 1 '^^c'" f^!/"" '^"'^'0 



At the pomts oi hiehest temperature — = oo , — ■ = co , — 



= 00 . 



•»-r '^I'V (ix^v, , dx"v.~, , 



Now — -; =: -rz H 



dx^ dx^ dx'^ 



and any values of x, y, z, which make either term in the second member 

 infinite, will make the whole member infinite, and insure a shooting 

 point for v as far as the direction x is concerned, wherever there is one 



for either v, or v.,. The like is true of y and z. If -^—^ and — - become 

 ^ - ^ dx^ dx^ 



CO with opposite signs, there will" be an ambiguity which can be got 



rid of by the determination of the indeterminate quantity (» — co ). 



Using the results of both (2) and (3), it is easy to arrive at an 

 absurdity in almost any case considered. For instance, heat to a con- 

 stant temperature all the points upon a circle, marked out upon a large 

 metal plate protected from surface radiation. Then not only will the 

 point at the centre of the circle soon become hotter than any point upon 

 the rim, in which case the heat must flow from within out, but it must 

 finally be infinitely hot, which is absurd. 



Our experiments have not proceeded so far that we can determine 



