152 LECTURE XVII. 



contracted or made cycloidal, and would lose 10 seconds by having the 

 vibration extended to an arc of four degrees. In order to avoid the friction 

 which would be occasioned by the motion of the pendulum on an axis, it is 

 usually suspended by a flexible spring which is wholly free from friction. 

 The elasticity of this spring adds a minute force to the power of gravitation 

 which acts on the pendulum, and this force must be considered, when the 

 length of a simple pendulum is compared with the frequency of its vibra- 

 tions. It does not, however, interfere with the equality of the vibrations 

 among each other ; for in all springs, Dr. Hooke's general law,* that the 

 force increases as the degree of flexure, is found for moderate oscillations 

 to be perfectly accurate ; such a force, therefore, accelerates the larger and 

 the smaller vibrations precisely in the same degree. But in balances, it is 

 desirable to have the velocity and the extent of the vibration as great as 

 possible, in order that the motion may be the less influenced by the ine- 

 qualities of the sustaining power ; and in large excursions, Dr. Hooke's 

 law is not so precisely true ; there must also necessarily be some inaccuracy 

 from the loss of a certain portion of the force in generating the momentum 

 of the spring itself, which, when the form is spiral, introduces great intricacy 

 into the calculation of the properties of the vibration. Yet it has been 

 found by experiment that a certain kngth may be determined for almost 

 every spring, which will afford vibrations either perfectly or veiy nearly 

 isochronous. In order that the weight or inertia of the spring may inter- 

 fere the less with the regularity of its motion, it is sometimes tapered and 

 made thinner at the extremity: it is now also usual in the best watches to 

 employ a spring coiled into a cylindrical form, like that of the spring of a 

 bell, of which the motion appears to be somewhat more regular than that 

 of a flat spiral. This was indeed the original construction, but was pro- 

 bably laid aside on account of the space which it required. The balance 

 springs are made of the finest steel, and the best are manufactured in this 

 country, although the French are said to have the art of making their main 

 springs of a better temper than ours. Sometimes the balance spring- is 

 made of an alloy of gold and copper ; these springs are very elastic, but 

 they are too liable to break. Mr. Earnshaw observes that the strength of 

 a spring always diminishes a little as it wears ; and endeavours to derive 

 a compensation for this diminution of strength, by employing a spring of 

 such a form, that the vibrations in small arcs may be a little more frequent 

 than in larger ones, in order that when the presence of dust and the 

 tenacity of the oil contract the extent of the vibrations, this contraction 

 may tend to produce an acceleration which compensates for the diminished 

 force of the spring. But it is perhaps more eligible to make every com- 

 pensation, as far as possible, independent of circumstances foreign to the 

 cause of the error. The strength of the spring is found to be less impaired 

 by use when it is hardened than when the steel is softer. It sometimes 

 happens, that from a sudden motion, or from some other accidental circum- 

 stance, the balance of a timekeeper may be thrown beyond the point at 

 which the pallets are impelled by the scape wheels, and the whole motioli 



* Hooke, De Potentia Restitutiva, 4to, Lond. 1678. This law was published by 

 him about the year 1660, in the form of an anagram. 



