One of the most brilliant exhibitions of skill in velocipedes-trianism that has ever taken place in this city or elsewhere, took place at Apollo Hall, corner of Twenty-eighth street and Broadway, a few evenings since, under the direction of the Pearsall Brothers. Dodworth's band was present, and the evolutions of the skillful riders present on the occasion were rendered more pleasing by the accompaniment of splendid music, for which this celebrated band is distinguished. The tournament opened by the entrance upon the floor of twenty-five of the most expert riders in the country, whose advent called forth immense applause, renewed as the graceful evolutions of the performers excited and delighted the admiring assembly. The affair was very select, and was attended by a large and fashionable concourse of ladies and gentleman. Nearly all the bicycles in popular favor were represented, but the most attractive feature of the evening was the performance of a sister of Messrs. Pearsall, on a beautiful" little ladies' velocipede, which has been appropriately called the " Peerless." This machine has low wheels, and is propelled by treadles connected with the cranks, so that a special dress is not required by the fair rider. It is altogether a most attractive design, and will, we think, speedily become a favorite with the fair bex. A two hundreddollar Pickering velocipede is offered by the Pearsalls, to be competed for the fastest time in a half mile at the Gymnacyclidium, on Thursday, the 15th inst. The machine is mounted with silver plate and ivory fittings, and is a gem. A challenge has been put forth by Mr. Frederick Hanlon, who offers to race any velocipedist of the United States for a thousand dollars a side and the championship. The race to take place in this city or Brooklyn, half mile heats, best two out of three. The time between the heats to be ten minutes. The party accepting the challenge to choose his own velocipede, the fore wheel of which shall not exceed 37 in., except it be a Demarest, in which case the fore-wheel shall not exceed 41 inches. The Herald says: " It is probable that a Brooklyn expert will accept Mr. Hanlon's $1,000 challenge, and that the race will be arranged to come off at the Empire City Kink." Mr. Stephen W. Smith has commenced a suit against Mr. Calvin Witty for alleged infringement upon patents originally granted to Philip W. McKenzie, of Jersey City, and subsequently assigned to Mr. Smith. The McKenzie invention was illustrated in these columns a few weeks ago. Much diversity in opinion, as to the proper dimensions of the velocipede wheels and cranks, has existed, but the favorite size seems to be from 30 to 36 inches for diameter of driving wheel, and 6 inches for length of cranks. We have seen larger ones, but we doubt that they will be much used so long as the bicycular form of velocipede is considered the best. Since writing the paragraph in regard to rubber tires for velocipedes, we have had submitted to us a number of plans for fastening them. To fasten them firmly has been the difficulty heretofore. Some of the plans proposed seem well adapted to meet the requirements of the case, but actual trial can alone demonstrate their value. We saw recently a bicycle propelled up the heavy grade from the Wall Street Ferry to the top of the Brooklyn Hights. We were too far away to ascertain the maker of the machine or the name of the rider* When we add that this grade is certainly not less than one foot in ten, our readers will appreciate the significance of this statement, with reference to the possibility of overcoming steep grades. The rider ascended the entire grade, certainly not much less than three hundred yards in length, using the flagged sidewalk as a way. The BrooMyn Union says, the fastest time yet made on a velocipede in this country, was that made by Messrs. Burroughs and Demarest, on the night of the third inst., on Demarest machines, with 45-inch and 41-inch driving wheels. The trials took place on the mammoth rink in Third avenue, and the machines which were ridden were the Demarest, Wood, Pickering, Mercer, and Monod, and the Union Hardware Company. Previous to the race the Tilton Brothers and the two Tildens did some bicycle gymnastics, and the display was much admired. We heard a suggestion made that the exhibition would be preferable if the two parties went in couples rather than in a quartette After the fancy riding came the races. The course was half a mile, three times the circuit of the hall, the center of the hall being marked off by rows of seats for exercise riding. Mr. Burroughs led off on a 45-inch Demarest, and he went round at a startling pace, making his first circuit in eighteen seconds, great time for the sixth of a mile. He, however, started too fast to keep up his pace, and he occupied 72 seconds in doing the entire distance. Darling was the next, and he made the half mile in 7l seconds on a 4l-inch Demarest. Young Hamburgh now tried in on a a 33-inch Union Company machine, and he made excellent time, coming-in in 85 seconds Mr. C. D. Demarest now got on a 41-inch Demarest machine, and he flew round the hall at a rapid pace, coming-in in 68f seconds! the fastest half-mile time on record A Mr. Weed then tried a 38-inch Pickering, "but it took him 90 seconds to go the half mile. G. Tilden then tried his skill on a 45-inch Wood machine, and he did his half mile in 76 seconds, his brother doing it in 83 . A rider named Capeless was the last, and he went round on a 854nch Monod in 84 seconds, and thus ended the trials.