S P I 



tre, and .r, y, % those of any point on the surface j then the general equa- 

 tion is 



(.r-.r') a + (y-y')* + (*-*') =r. 



If the origin be at the centre, x', y' t and % each o, and the equation 

 becomes 



SPHERICAL excess. 



Spherical excess in Trigonometry is the excessfof the sum of the three 

 angles of any spherical A above two right angles. Now in surveying a 

 country where the sides of the A's are usually 14 or 15 miles each, the 

 spherical excess, with a fine instrument, is plainly discernable ; and in 

 strict accuracy the sides of the A's ought to be calculated by the rules of 

 spherical Trigonometry, which would be a most tedious process, where 

 many hundreds of such operations are to be performed. Legendre has 

 therefore furnished us with the following rule, which combines sufficient 

 exactness, with all the conciseness that can be expected, viz. : 



A spherical A being proposed, of which the sides are very small with 

 regard to the radius of the sphere, if from each of its angles one-third of 

 the excess of the sum of its three /'s above two right /'s be subtracted, 

 the angles so diminished may be taken for the /'s of a rectilineal A, the 

 sides of which are equal in length to those of the proposed spherical tri. 

 angle. 



SPIRALS. (Higman, VinceJ 

 1. Spirals y Equations to. 



In the spiral of Archimedes, let r =r rad. vect. 9 = /. traced out by r ; 

 then 



r = - . 6, or r a 8 : if a = - . 



2 * ' 2* 



In the reciprocal or hyperbolic spiral, 

 _ a 



In the logarithmic spiral, 



9 



r a. 

 In the lituus, 



The spiral of Archimedes, the reciprocal spiral, and the lituus are par. 

 ticular cases of the equation r a ffl 1 . 

 270 



