102 Mr DAVIES on the Nature of the Hour-Lines 



1. Let cot x cot I be greater than 1, 

 and have the sign plus. It is plain 

 that the denominator is always posi- 

 tive and always finite, and hence v 

 must have values oscillating between 

 finite limits. 



If, on the contrary, cotx cot I be greater than 1, but marked minus, the 

 denominator will always be negative and finite, and therefore oscillate be- 

 tween finite limits. The value of v, then, also possesses the same character, 

 and differs from the other case only inasmuch as it is measured upon the 

 radius sector in the reverse direction, or negative side of the pole of co- 

 ordinates. The figures are in both cases exactly alike, then, in form, but 

 reversed in position. The annexed sketch will give an idea of its character. 

 The details of its course are easily laid down by enumerating the changes 

 that result from the gradual change in the value of m L, and are too simple 

 and obvious to need recapitulation, especially as they resemble the investi- 

 gation contained in II. almost to identity. 



2. Let cot x cot I 1 ; then when wL = (p + 1) x 90, the curves run 

 out to infinity, and v becomes an assymptote. They then repeat the same 

 system of changes through the next L, and so on without end. They have 

 for the least value of v, 



v = - cosec X sec X : 

 2 



and where the curve crosses the equator, it is 



v = a cosec X sec X. 



If cot * cot I = 1, the same curves and changes, but in a reverse 

 order, present themselves. By comparing the sketch here given with Mr 

 CADELL'S projection on the equatorial tangent plane, it will be seen how 

 much alike the curves are in the general form ; but by comparing the equa- 

 tions of XVIII. 2. and that which will be given in XXV., a striking dif- 



