DEPARTMENT OF PUKE MATHEMATICS. 485 



If a = f, the Anemoids are curves of pursuit. 

 The path of P relative to Q is 



r/c = — e-'j(cos « + /x sin 6) 

 where r, 6 are polar co-ordinates referred to the storm-centre as origin, and 

 t = p^ when /i<cos a 



If = 0, the path is a conic section for all values of /x, and if ^ = I, the path is 

 a parabola for all values of a. 



The interpretation of the equations shows that Anemoids belong to four 

 classes : 



1. Spirals: /i<cos a. The coils intersect, touch, or fall each within the pre- 

 ceding, according as /x > = <sin a. 



2. Curves with a stop-point: /x= or >cos a, but ^l<\. The air-particle falls 

 into the storm-centre at the stop-point. It is shown geometrically that the air- 

 particle proceeds thence along the storm-path. 



3. Curves with an asymptote: /i = l. The asymptote is parallel to the storm- 

 path. 



4. Loops: fi>l. 



These classes may be further subdivided according to the value of a. 



The trajectories traced from observation by Dr. Shaw and Mr. Lempfert 

 present in several cases a considerable resemblance to portions of Anemoids. When 

 the storm-centre moves in a straight line, the air-trajectories may be looked upon 

 as curves derived from Anemoids by supposing /x and a to vary ; if ji and a vary 

 very slightly the Anemoids may be regarded as a first approximation to the 

 trajectories, 



3. On Residues of Hyper-even Numbers. 

 By Lieut. -Colonel Allan Cunningham, E.E. 



Let 2^1= 1 (mod.w), 2^'^e: 1 (mod. .?,), . . . 2^'*'^— 1 (mod, v,), 

 and li = 2«-,i, i, = 2'"lv . ■ . l,+i = Si"'':\'7r+i 



where m, ijj, t/^ . . . are all odd ; and $i, |j, ... are all minima. 



Then ^j, ^o, &c. are the Haupt-exponeitts of 2 to the moduli ??i, r;j, &c. The 

 T) diminish at each step, until eventually (for all moduli m alike) i7,._i = some 

 Fermat's number, ^;=2''^ '7)=1, |i+i = l. All moduli (m), which have the same 

 |,, have the same train of ^r, ^in 



Let E„, „ = 2«, El, „ = 2E<'.», . . . E,, i,„ = 2'=.-, ... 



These are Hyper -even Numbers. Let E,., „ be the residue of E,., „ to any odd 

 modulus m. The residue system of the rth order, i.e., of E^, „, E,., j, . , . Er, „ 

 (with the same r) to mod. m consists of txuo parts : — 



(1) A non-recurrent part of, say, h' = hy+ 1 terms (Rr,o, Rr, i • • • Rr,*)- 



(2) A recurring cycle of, say, k^ terms (R;,+_j, K;.+2, . . . R^+a.). Here A:r= ^r+i> 

 and h'r is the minimicm satisfying the inequalities (of which the first two or three 

 usually involve all), 



/(',.<): Or, 2 '*'r<j:a,._i. El,,/ <Co,._2 . . . 



In passing from one system E,., „ to one of higher order (r + x) the cycle- 

 length ('^v = |r+i) decreases rapidly with increase of r, until at length the cycle is 

 reduced to a shigle term (R,..;,^l = constant), the order (;•) being found from 

 ^r = ^,+j = l ; also for all higher orders (r ■{■.%•) the cycle Is of one term oply, and 



