ON TERRESTRIAL MAGNETISM AND ATMOSPHERIC ELECTRICITY. 753 



whicli -will allow the above integral to be expressed as a finite sum of integrals 

 havinsr one of the forms : 



^o 



Jo 



V cos p6 cos <rX d6d\, 

 or 



V sin 2^6 cos o-X ddd\. 



J J 



In other words, if V is expressed in a series the terras of which are of the form 

 cos pB cos o-X or sin jt)^ cos o-X, and if equations are obtained once for all to give 

 cospd or sin pd in a series of tesseral harmonics, the problem of expansion is solved, 

 though the independence of the higher and lower terms is not necessarily secured. 

 The first step of the procedure which is common to all methods consists in 

 expressing V in the form 



« - V = F.j + F, cos X + F., cos 2X + . . . 1 /■■ ^ 



+ F/ sin X + F/ sin 2X + . . . /■ * * * ^^ 



where F and F' are functions of the colatitude. If V is given discontinuously ; as, 

 e.ff., if it is known at the points of intersection of a number of latitude and longitude 

 circles, each latitude circle will give an equation of the above kind. If the number 

 of such equations is sufficient, the values of F and F' maybe plotted in terms of 6, 

 and by calculation or mechanical integration we may obtain the values of F and F' 

 either as a series 



Fa = »„ + a^ cos 6 + «., cos 2^ + . . • . (2) 



or in the form 



F„ = 6j sin 6 + b.,s\n2d+ (3) 



Only one of these forms is useful for our purpose, as I proceed to show. 



If (7 be even, T° may be expressed in the form of a finite series of the form 



T,," = (',, cos uO + a„^2 cos (n - 2)6 + . . . 



Hence, if o- be even 



f T,," cos ;j5rf5 = ifp>M; 



k Jo 



and the integral will also vanish i{ p + n is an odd number. It follows that for <r 

 even 



C + 1 f 9 



T„ sin pddix = i T,: [cos ( ;; + 1) (9 - cos (^j - 1) ^] dd 



•' -1 ' •' 



= if « > M or if (p + n) is an even number, 

 t .-1 ^ 



It follows that (o- even) we may obtain a number of series in which the sines 

 of multiples of 6 are expressed, as in the following scheme, where the numerical 

 coefficients are left out for the sake of simplicity : — 



sin 5 =T/ + T/^.,-f- • • 



sin 2^ =T/,i + TA3+ • 



sin<r^ =T/,i + T/+3+ . . . 



sin(o-+l)<9= T/+2+ . . . 



sinV^ = T^,i+T/+3 + 



(^) 



Whenever p is smaller than o- the series begins with the term T/or T„a» 

 according as p be odd br even ; but when p is larger than a- the series begins with 



Ttr 

 p + 1 • 



1898. 3 c 



