306 G. H. KNIBBS. 



density of the shading on the planes XY, XY denoting the 

 intensity, and the heavy spiral line the locus of one of its maxima. 1 

 In a refracting seolotropic region, with a suitable distribution of 

 density, the curves of minimum path, for motion affected by the 

 seolotropy, 2 may be regarded as the straight lines of the space ; 

 space of this kind is in general tortuous space. 



31. The total intensity-volume of ceolotropic space. — ^Eolotropic 

 space measured with respect to volume only, is of course, not 

 differentiated from ordinary space; if however its intensity is 

 taken into account, that is if an element thereof be 



8Pi = / xyiw ... 8x 8y8z8w (28) 



in which Fj denotes the intensity-volume I m V, or the mean intensity 

 multiplied by the spatial volume — this is no longer true. It must 

 suffice to take a single illustration of an elementary character. 

 Suppose that the linear intensity in a spherical space surrounding 

 the point O, is some such simple function of the distance r there- 

 from, as 



I^l+ar 11 (29) 



where a is any finite number, 3 and n is positive or negative. The 

 intensity-volume is obviously 



V. = 471-/(1 + ar n )r 2 dr = f irr 3 + 4:7rafr 2 + n dr (30) 



that is to say when n is - 3 



V l= $ 7rr 3 (l+3a l0g 3 r ) (31) 



and in all other cases 



V { = t7rr 3 (l + 3a -JL) (31a) 



o + nj 



1 The distribution of an electrified powder on an electrified resinous 

 cake, in Lichtenberg's experiments — Chladni's sand-figures on a vibrating 

 plate, on Strehlke's or Faraday's modification of this experiment, the 

 distribution of liquid spherules in a vibrating bell partially filled with 

 liquid in Melde's experiment — maybe taken as illustrating the actual 

 distribution of intensity. 



2 In the atmosphere a ray of light is bent toward the normal to the 

 surface, when the pressure and temperature vary uniformly upwards. 

 Measured by a time-unit this curve is the shortest path, owing to the 

 irregular distributions of temperature and pressure the actual path of a 

 star's light is always more or less tortuous. 



3 Which we shall see may be positive or negative. For simplicity's 

 sake we may suppose the intensity at O to be unity if n is positive. If 

 negative it will be infinity. 



