(;.-,n 



SLOW ON THE VARIATIONS IN 



On the Method* of ascertaining the Numerator and Denominator of Fractions 



representing generating spirals of all series. 



In order to find the numerator and denominator of the fraction representing a 

 LTt'iicratinu" spiral, where the leaves or scales are so crowded that it cannot be readily 

 discovered by inspection alone, the usual rules are as follows : — Either to affix*, first of 

 .•ill, tin- proper numbers to each scale, and then observe that which falls upon the scale 

 vertically over the first, and which, lessened by 1, gives the denominator; while, to find 

 the numerator, the axis must be allowed to revolve, when the number of revolutions 

 made in passing from No. 1 to the scale over it will give it. Or, if we take the common 

 didei nces corn-- ponding to the two secondary spirals which pass through the scale 

 leleeted as No. 1, and also the scales nearest to, but immediately below, the scale ver- 

 tically over the first, the sum of these common differences supplies the denominator, and 

 the smaller of them the numerator. 



The former of the two methods will apply to any spiral of any series. The latter rule, 

 though of application in the primary, fails to give the numerator for any divergence 

 other than those included in that series; but if it be remembered that the numerators 

 ire respectively the same for the corresponding fractions of every series, no great 

 difficulty will be met with in affixing the right one to any denominator. This arises 

 from the fact that the number of coils in a cycle is the same for all such fractions, the 

 real difference being in the number of leaves in a coil. Thus #. -A-. A, are the third 



83 115 143 



feai ns in the primary, secondary, and tertiary series, in which each coil of a cycle has 



3, 4, 5 I i?ea respectively. 



The fact of each coil having the same number of leaves, whatever arrangement be 



taken (provided it be from one and the same series), appears to be a very important 



prmciple, and hitherto, I believe, overlooked. Yet it is mainlyf upon this fact that all 



dculattons are really based. That this is the case will be understood from the fact that 



it any spiral of the primary scries, by a diminution of the angular divergence, were to 



Q^^. V^WV,.^ 



dary 



pc K i four loaves in a single coil, it would become a member of the „».. _ 

 S.nularly, u a spiral of the secondary series shall possess five leaves in any of its coils 

 v ill pass into the tertiary series. 



On cerl„in Belation, bctueen the Fractions of the several Numerical Series. 



It has be,,, ;,l ady noticed that the sum of the numerator and denominator of any 

 Zl ,L°7r «"'" 'I t0rm : f the i denomi - tOT of the corresponding fraction in the next 



sen. , thus, if they be written down as follows, 



Primary series ...1123 5 8 130 

 Secondary series . . Ill jl"j 8 1 3 



rn 4.- . 3 ' 4 ' 7 ' llJ T¥> 29, if, &C, 



Tertiary series . . . 1 i i jl s s 13 p 



* It 13 assumed that the reador u fc»«,a;«. - A u «*. . 



t It must he remembe J ttl as Zt t T ^ *"* * ^^ text - bo ° k8 ' 



for its position. vi z . from 1 30° to 180° i T T ^ *"* ° f & 8piral ° f the P rimai 7 8eries has a ran S e of 60 ° 

 inclusively, ther, mint be some « inna't 'V'd^ ' ^ ^^ divei S ences of tha * *™ lie hetween \ and £ 



( voml , which gives these second leaves a ten!^ ^f*" (f °' Want ° f a better ^P^ssion till the cause be dis- 

 the sucoe«ive convergent* of the priman- seriesThe H 7 T ^ **** WitWn that range ' and corres P onding t0 



